Starting with zza ni's numbers where the base magic cost is +10,000gp over the masterwork armor, there are three possibilities:
It is +1 armor with +9,000 gp of addons,
Upgrading +1 armor to +2 costs +3,000 gp. Upgrading +2 to +3 costs +5,000 gp. Upgrading +3 to +4 costs +7,000 gp.
If the player and DM are happy with the +7,000 number, then I would run with that number.
But looking at the benefit, I just don't see it being the equivalent of a +2 bonus. 3d8+5 healing once per day? A standard magic item that does that would cost:
Spell Level (3) x Caster Level (5) x 1,800 /5 (one use per day) x 1.5 (multiple different ability penalty) = 8,100 gp. That is really close to the +9,000 number from the first line.
Based on that, as DM, I would offer the 3,000 gp number for the upgrade. As a player, I would accept the 5,000 gp number.
I agree with Dasrak - that seems like a pretty balanced party to begin with. You should be able to play what you want to play without worrying too much.
Since you said that you want to shore up the party, I do have some thoughts to share.
With your experience and four new players, playing a buffer character may make their game more enjoyable for them. I would have recommended a Bard, except you already have one. The Skald class is very similar, but the buffs from the two classes can stack for the entire party. Pick a different set of skills to focus on, or just tag-team with the bard on important things like Diplomacy. Pick "Urban Skald", and the Magus and Warpriest will love you.
If you do not want to occupy so much turf of another character for the entire campaign, you can decide to switch-hit with a class like the Medium. You need another fighter? Today he is a champion. You need emergency healing, that day he is a Hierophant. You need a class skill that no one wants on a permanent basis (Profession: Barrister)? Then he can channel a trickster. The downside to the class is how complicated it can be.
The last of my trifecta of support characters is the Unsworn Shaman. It can play a lot like a wizard in being very flexible when using a Lore spirit and the Arcane Enlightenment hex to pick and choose which spells. The plus side is you do not have to find the spell in a book! The down side is that you need decent CHA, WIS and INT scores for a character. That, and the character can get stuck playing as a cleric, which you said that you did not want.
The above was my first take. My second take is that your party is a little short on ranged melee. You have a blaster sorcerer, but an archer would fill in some gaps that the other characters could not easily do. Hunter? Zen Archer Monk? Ranger?
But then I started thinking about the current party's makeup. One full caster and three 2/3 casters. That suggests your sweet spot where you do not get left behind and do not outshine everyone else would be another 2/3 caster. So, back to Hunter or Skald. Or maybe a Questioner Investigator.
Personally? I would pick a dexterity-focused Halfling Swashbuckler (Inspired Blade) 1/Medium 16 that focuses on Dex and uses a Rapier. (Favored class: Medium, 15 favored class bonuses into tha Seance Boon).
Doing that delays the decision about how to bolster the party to each game session you play. Champion, Marshal and Trickster spirits all use the same ability score. Rest a day, and switch to what you need. Need a caster? He may not be a true 2/3 caster, but in a pinch he can pick up Arcane or Hierophant and mimic one when needed. With a dex focus, he can put a few points into the mental skill (CHA) needed to cast.
And some of the spells he gets as a half caster are pretty solid. Teleport as a fourth level spell? No one else in the party will have that.
#45 Need part time workers to distribute seasonal flyers and menus for Chez Miss Shelyn theatre, restaurant and tavern. Work as little as one day every 3 months!
When asked, the owner wants those menus and flyers in the hands of every noble, guild master, business owner, crime boss, war lord that can be reached ... anyone with money or power!
The theatre is real and is professionally run by a bard. The food and drink are very good, and does rotate the menu every 3 months.
The trick? The owner has put an image of his face on every menu and flyer. He is a spymaster that casts Ears of the City to learn who is interesting for the day, and then casts Enter Image to keep tabs on those interesting people.
That ... is an interesting question.
First things, though: Evangelist 5 is only +3 BAB.
Back in 2011, Sean Reynolds said that the Eidolon's BAB replaces just the Summoner's BAB bonus, so in your Fighter 5/Synthesist 5 example:
When the Eidolon has not been summoned, +8 (Fighter +5, Summoner +3)
When things get interesting is the interaction with the Evangelist class, since it can modify the Eidolon, which is a class feature of the summoner.
Things are simple for a Synthesist 5/Evangelist 5 that has not summoned an Eidolon, as the base attack bonus for the classes add up to +6 (Summonor +3 and Evangelist's +3).
Equally simple for the same character that has summoned the Eidolon but has not performed the daily obedience, the base attack bonus is +7 (Eidolon's +4 and Evangelist's +3).
But with both the obedience performed and the Eidolon summoned ... oh dear. The Eidolon is now based on a 9th level summoner, so 7HD and +7 BAB. If that +7 replaces only the summoner's +3 while leaving the Evangelist's +3 intact, then is that +10 BAB?
Hmmm. I think that breaks the spirit of the rules, so in my campaign, I would not allow that. Instead, I would have the Eidolon's BAB replace the BAB from the class levels that provide the Eidolon bonus (all three summoner class levels, 4 of the 5 Evangelist class levels), so the summoned+obedience character would have a net BAB of +7 (7HD Eidolon +7, +0 from first level of Evangelist).
The reason I would rule that way is the general principle that "bonuses of the same type from the same source do not stack". The Evangelist class levels are the source of +3 BAB from the class itself, and the source of the +3 BAB from the obedience-only Eidolon HD.
But how bad would it be to rule the other way?
Synthesist 5/Evangelist 9, with Obedience performed and Eidolon summoned. The Eidolon will have 10 HD, so a BAB of +10. The Evangelist class has a BAB of +6. Allowing those to add together would give a BAB of +16 which is enough for 4 iterative attacks ... two levels before a single class fighter can do the same.
And I'm right back to "Nope".
Just my $.02
Are there any spells that you wished you were able to cast, but simply did not have enough slots for?
If so, consider the Page of Spell Knowledge: https://www.d20pfsrd.com/magic-items/wondrous-items/m-p/page-of-spell-knowl edge/
1,000 gp for one more first level sorcerer spell you can cast,
If your character is never being attacked with a weapon, then increasing the ring of protection to +2 seems a waste. But if your character is in combat every few seconds, then ... mayyyybe that extra +1 to AC?