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Ler the advisors tell the PCs about the powers and spells of the BBEG and then let the PCs decide what their strategy will be.
Make it so the advidors can also arrange for buying any objects the PCs would want (within reason).
Sit back and enjoy the players' imagination going wild.
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Level +3 enemies are not going to be easy. That is at the maximum edge of what a well-prepared, fully rested group of tactical experts with optimal characters can handle.
I'm not sure what help I can give for this, since I haven't ever actually played at that level. I have heard that at higher levels that the CR+3 enemy won't be one-round-KO'ing PCs (like they often do at low to mid levels) - or at least it will be less likely. Though as you point out, this particular enemy has some crowd-control spells that it is going to be landing with high probability and criticals with more than low probability.
Unless it's heightened, baleful polymorph shouldn't be a problem. Preventing other spells is simple, in theory. Put silence 4th on somebody (or multiple somebodies), cast anything that inflicts slowed 1 (or maze) and have a caster grab a bow, load up on true strikes, and pump bola ammo into it to keep it prone and open to the barbarian's AoO on stand.
Optionally, provoke combat by sending in an illusory party to soak up an opening salvo while you sit further back and drink some potions like mistform elixers or buff up. Don't need to worry about spells that aren't targeting you after all.
Prepare some snares beforehand too. In a pinch, you can run out and maybe lead out through some traps to chip it down if it's injured enough.
There are two other major issues though. First is going to be damage in that they have none. Maybe pack a some manifold missile wands to get some automatic force damage going. The second is going to be enemy saves. Between the level gap, the lowest being moderate reflex and the +1 vs magic, your casters are going to struggle to land anything and you really need to hope it doesn't roll much better than average because it will crit succeed often. Illusions are your friend here since even if they're easily seen through, it's going to lose actions to do it. Maze is also your friend as it will 100% consume at least one action to undo.
If you're in a cave/lair, this should be mostly straightforward aside from its ability to burrow. If you're outside, I highly doubt this party is optimized enough to handle it without some truly bad tactics on the dragon's part. If it burrows, at least it's moving slowly. Make sure everyone has darkvision and consider jumping down after it if they don't have any other way to deal with it.
Unlike breithauptclan, I can confidently say that a good party can handle all the way up to CR+5, especially at this level and with prep time. That party is not this one though. This is going to be brutal for your players.
Level +3 enemies are not going to be easy. That is at the maximum edge of what a well-prepared, fully rested group of tactical experts with optimal characters can handle.
I'm not sure what help I can give for this, since I haven't ever actually played at that level. I have heard that at higher levels that the CR+3 enemy won't be one-round-KO'ing PCs (like they often do at low to mid levels) - or at least it will be less likely. Though as you point out, this particular enemy has some crowd-control spells that it is going to be landing with high probability and criticals with more than low probability.
In contrast, due to running a 7-player party, I am accustomed to the party encountering a level+3 opponent because a larger party needs a larger challenge. And they have played through 15th level.
The high-level monsters have heaps of hit points. If the party is built for massive damage, then they could destroy that heap quickly, but my party is build for defense and tactics. The high AC 42 of the ancient blue dragon will be frustrating. The players will feel like they are nibbling at the edges of that heap of hit points due to frequent misses. For further frustration, the attack bonus of animial companions does not increase enough at high level, so the bear will almost always miss.
The battle will feel much better if they can debuff the dragon's AC. A rogue in my party took Precise Debilitations feat 10, which can make the monster flat-footed to everybody, and all the party members were grateful for the -2 penalty to high-level ACs. When a PC can ordinarily hit only on a natural 17 or higher, hitting on a natural 15 or higher is 50% more successes.
Spells likewise fail more against a level+3 creature's saving throws, so expect a lot of wasted spells. Fortunately, Paizo designed several spells to give a one-round debuff on a successful save, and I already pointed out that debuffs feel great. So coordinate the spells with the martials so that they can take advantage of the consolation-prize debuff.
The worst part about nibbling away at the high-level enemy is that that enemy is dealing heavy damage to the PCs while they nibble. A party facing several opponents ends up with the damage spread across several PCs, but a single opponents automatically concentrates their damage in one turn against a single PC. If that PC is the target on the second turn, then that PC could be knocked unconscious and dying, which seriously cripples the party. The party has to avoid that happening, and the GM should fudge a little in their favor, "The dragon decides that the cleric healing the three-quarters-dead barbarian is a nuisance, so it attacks the cleric this turn."
This is where the champion in my party shines. She protects people. When a level+3 monster attacks a party member, she had the foresight to stand 15 feet away and protect the person with Liberating Step. If the monster attacks a party farther away--which is likely with a highly-mobile ancient blue dragon--she resorts to Champion's Sacrifice, absorbing all the damage herself. Could markrivett include a champion NPC?
My party also typically uses terrain to protect themselves. For example, they might attack from the forest, so that an injured PCs could retreat and hide in the underbrush to safely drink a healing potion. Using terrain is less effective against an ancient blue dragon with Speed 50 feet, burrow 30 feet, fly 200 feet and built-in area of effect damage. Blue dragons even have the hallucinatory terrain spell to alter the terrain. The terrain will probably favor the dragon.
The description sounds like they know that they will face an ancient blue dragon. Thus, let them make Recall Knowledge checks during downtime, based on the local's accounts of the dragon. "I saw the dragon breath fire in a long straight line, except the fire was blue and sparky rather than hot and yellow." If they fail a check, then let them roll again against the DC of an adult blue dragon, which technically is a distinct creature, so some indirect information. Advance information will help the party prepare, and they will not have to spend combat actions on Recall Knowledge.
The good news about an ancient blue dragon is that they are haughty and love to fool people. It will probably love to show off, watching people quake in its Frightful Presence, casting hallucinatory terrain so that they run in circles, and taunting them with bad illusionary-creature parodies of themselves. It might accept a few turns of damage before it gets serious about battling the party, and the fright from Frightful Presence will have worn off.
In contrast, due to running a 7-player party, I am accustomed to the party encountering a level+3 opponent because a larger party needs a larger challenge. And they have played through 15th level.
I have been in combats similar to this more than a few times. It isn't very much fun from the player side. Because this:
The worst part about nibbling away at the high-level enemy is that that enemy is dealing heavy damage to the PCs while they nibble. A party facing several opponents ends up with the damage spread across several PCs, but a single opponents automatically concentrates their damage in one turn against a single PC. If that PC is the target on the second turn, then that PC could be knocked unconscious and dying, which seriously cripples the party.
happens. A lot. And at lower levels, that KO result often happens on the first turn that one of the player characters draws aggro. We were in a PC level 6 vs enemy level 9 battle and had two different characters go from full HP to KO in one round.
Also, sure, the party may win the fight eventually. But any one particular player will not feel very effective. The Barbarian that does Stride, Miss, Miss for three rounds in a row before being KO'ed doesn't feel like they contributed.
In contrast, due to running a 7-player party, I am accustomed to the party encountering a level+3 opponent because a larger party needs a larger challenge. And they have played through 15th level.I have been in combats similar to this more than a few times. It isn't very much fun from the player side. Because this:
The worst part about nibbling away at the high-level enemy is that that enemy is dealing heavy damage to the PCs while they nibble. A party facing several opponents ends up with the damage spread across several PCs, but a single opponents automatically concentrates their damage in one turn against a single PC. If that PC is the target on the second turn, then that PC could be knocked unconscious and dying, which seriously cripples the party.happens. A lot. And at lower levels, that KO result often happens on the first turn that one of the player characters draws aggro. We were in a PC level 6 vs enemy level 9 battle and had two different characters go from full HP to KO in one round.
My players have fun in these level+3 encounters because they have a strong emphasis on keeping each other on their feet via teamwork. I mentioned the champion. The druid and the sorcerer both cast heal spells from their mid-level spell slots. The druid and the monk both studied Battle Medicine. They position themselves to protect injured party members. And between encounters they take time to Treat Wounds back to full hit points.
Over 19 levels they have had only three knockouts and zero deaths.
Also, sure, the party may win the fight eventually. But any one particular player will not feel very effective. The Barbarian that does Stride, Miss, Miss for three rounds in a row before being KO'ed doesn't feel like they contributed.
That happened to me, rather to my GMPC, back in February. I have a crossover between the characters of our previous Iron Gods campaign and the current PCs in our Ironfang Invasion campaign, mostly involving the rebuilt android Casandalee, who calls herself Bonnie Cas. The next crossover character was my GMPC bloodrager Val Baine (PF1 Bloodrager Val Baine Converted to PF2). 18th-level bloodrager Val Baine and 16th-level swashbuckler Gossamer accompanied the 17th-level party to their battle against Queen Arlantia, final boss of Prisoners of the Blight. I had redesigned Queen Arlantia as a 20th-level witch (thus a level+3 opponent) and given her a 15th-level Marrmora, a 16th-level Vilderavn, and a 19th-level Vilderavn Herald as henchmen. That's an encounter budget of 250 xp, but for a 7-member party a Severe-Threat encounter is 210 xp, so I added the two NPCs allies.
Val's strongest attack was a single shot with her pistol loaded with Elemental Ammunition. She missed. Then the two vilderavns flanked her in mutual melee combat and dealt 260 damage to her. Val was down to about 25 hp and losing about 10 hp a turn from cursed persistent bleed damage. She would have taken more damage, but the champion belatedly noticed that she was in trouble (Val was not yelling for help like the PCs do) and Liberating Stepped her out of the flank. Then the sorcerer cast Regenerate on her to keep her hit points up despite the bleeding curse. Val guzzled her healing potions to get battle ready again, but by then, the fight was over.
Val's contribution to the fight was serving as a punching bag to distract the vilderavns while the party focused on Queen Arlantia. That role was embarrassing, but it did let the PCs earn more glory, which is the purpose of an NPC ally.
Where is the battle taking place? Can they control the battlefield?
Can your barbarian or druid trip? The druid pet is useless unless the druid can turn into dragon form, close the gap, and trip the dragon out of the air. Fortunately dragons are fairly easy to trip, so if you have Titan Wrestler with maxed out Athletics on a druid in dragon form with mobility to match the dragon, you can close the distance and trip them setting up the dragon for a Sudden Charge hammer by the barbarian.
Cleric is going to have focus heavy on healing while maybe summoning a celestial or fiend with magic missile ability. A Meladaemon is ideal because they can do a 2 action magic missile every turn for 1 action by the cleric. That is a consistent 6d4+6 attack for one action up to 120 feet. I hope they are a neutral cleric or at least a neutral deity, so they can cast an evil spell.
Wizard wants to focus on magic missile, invis, and reflex based damage spells. Try to land some hammer spells at range. Try to stay unseeable.
Big bonus for you is you have a lot of healing with the cleric and it's going to be 12 actions versus 3 actions.
Barb should focus on tripping if possible to keep the dragon spending actions to stay mobile.
It would be nice to know what the battlefield looks like. If you can seal off the dragon from directly attacking you while still hammering it from range with maybe wall spells, you could set up a real advantageous position.
If they are in a throne room, doesn't sound like the dragon will be able to use its mobility much to its advantage. That is good for your party.
I can give you some basic ideas groups use on bosses:
1. Trip the boss. If someone has maxed athletics and a good str, trip the boss. Keeps them from flying and forces them to stand up. Anyone can trip if their skill is good enough.
2. Have every caster that can stick a slow on the dragon at least once a round. Slow works for 1 round even on a fail. If you manage to stick one for a minute, game is over.
If you can trip and slow the target, they are dead, dead, dead.
3. Make sure the casters are not in range to get damaged by melee. Spread the party so only the barbarian and the pet with maybe the druid if wild-shaped are getting damaged.
This allows the clerics to focus heal and erase damage on a round by round basis. Clerics have a lot of heals and can erase a lot of damage.
4. Pre-buff the barbarian or anyone using attack rolls with heroism and electrical resistance.
If the barbarian has a good-one handed weapon or room for a shifting rune, shift his weapon to a one handed weapon and give him a free hand so he can trip the dragon.
Hopefully the barbarian is giant instinct as that will allow him to trip if doesn't have Titan Wrestler.
5. Group haste everyone using a 7th level slot.
6. You don't have synesthesia, so scour for other good debuffs to use. Phantasmal Killer is a nice one given it has a debuff and some damage even on a successful save.
7. Don't forget moment of renewal. It is a powerful heal spell.
8. Make sure you have a way to deal with invisibility and other common defenses.
9. Check your summons to see if anything does something useful your party can't do.
You should be able to win even against a CR18. Your action advantage and huge healing pool is a pretty big advantage in this encounter.