You have an unclosed paranthetical statement in your first paragraph. I’m having a hard time understanding the situation you describe as a result.
I don’t know of any way to parlay 10k gold into 7AC. Maybe 1 or 2 or even 3 (+1 enhancement on your armour and a ring of protection +2 costs 10k) depending on what all you’re currently wearing, but not 7. There comes a point where AC is simply outstripped by to hit bonuses. It sounds like you might be there. Have you considered looking into ways to add miss chances instead?
Or you could just not PVP, call me crazy.
Is this Ranger melee or ranged? You could just fly away from your problems if he's melee, or throw a Tanglefoot Bag at him and walk away. If he's ranged, and therefore Dex-based, the Tanglefoot Bag will lower his attack bonus by 4 (even if he makes the Reflex save, he's still Entangled).
If you follow Pizza Lord's advice, you could buy a Cracked Vibrant Purple Prism ioun stone for 2000 gp. This ioun stone would let you store the Shield spell in it, so you won't need to equip the Shield wand/scroll. This would also let you make the UMD roll outside of combat.
Fighting defensively can up your AC a bit more (2-3 depending on acrobatics skill). If you're throwing bombs you may be able to take the -4 attack penalty.
So it’s the player that has the problem with magic users not their character? I’m assuming you’re ok with pvp so why not look to going on the offence? It doesn’t matter how hard they hit if they’re unconscious.
I personally would try not to engage with them if they want to go down the pvp road ( but a small part of me would want to make a magic user that could take him out). There’s plenty of ways to rp two characters as not getting along without killing each other, derailing the campaign or ruining the fun for anyone else.
So someone suggested the shield spell - It's on the Alchemist extract list so you can use an extract and have it last 8 minutes.
3 ranks in acrobatics means when you fight defensively you get +3AC for -4 to hit.
That's +7AC without spending any gold ... go team! (Or you can spend 750gp on a wand of shield - Alchemists can use wands without UMD).
If you don't have 3 spare skill-points you're only at +6, you'll have to somehow get +1AC with your remaining 9250gp.
You can also get the blur spell. This is useful for if/when the ranger gets past your ac as there’s still a chance for the attack to miss.
The other option is to not play with jerks.
Yeah this is the best advice, talk about it out of character.
If you've already done that and the other character is being a jerk, remember he made his character first so maybe try to work with him to make something that won't cause a problem.
If you've already talked to him and you both like the idea of having this inter-character conflict then go ahead and make your character, it's worth making your character hard for him to kill, but don't try to kill him. If you end up specifically building a character to kill his WITHOUT his permission then you're kinda being a jerk.
Assuming everything's above board and everyone's happy, there's also a +1 size bonus to AC, so either be a small race or use Reduce Person (or both) and you can get better AC.
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Alchemical Allocation+ potion of barkskin, Extract/wand of Reduce Person, extract/wand of shield. Together thats +8 AC. Also, what's keeping you from using a Buckler?
*edit: I was thinking Barkskin was a 3rd level extract for Alchs. Just use an extract of Barkskin for a +3 mat A.C. and you're at a +9 total
While under the affects of a transmutation spell the Scale and Skin feat increases your natural armour bonus by +1 (+2 if the caster level is 10 or higher).
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Anyone who plays a character that restricts other player's choices for characters without a previous diacussion with the other players is a jerk.
You said you don't have room for a shield. You do if you give up a discovery:
https://www.d20pfsrd.com/classes/base-classes/alchemist/discoveries/paizo-a lchemist-discoveries/vestigial-arm-ex/
More generally, I don't think that trying to match AC vs. a full-BAB characters' to-hit is the best idea. If you're already considering relying on buffs to see you through the fight, I would instead suggest drinking displacement or protection from arrows extracts.
DRD, technically, since that doesn't 'give any extra attacks', it shouldn't let me attack with my natural hands and still use a shield. I'd have to run it by my GM, though.
To the best of my knowledge, using a shield for its AC bonus does not fall afoul of the "extra attacks" clause in vestigial arm. As long as you don't use it to shield bash you're golden.
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If you are seventh level or higher this might work. Alchemical Allocation to not use the potion up. The class ability enhance potion to make the potions your level (retraining maybe?) and amplify elixer to extend the potion. You cannot extend extracts.
Ironskin lasts a minute a level. If you are willing to put up with that there is an advantage with barkskin, it lasts ten minutes a level. Iron skin is personal and cannot be made into potion form. Barkskin can be made into potion form. Look, if you are sixth level or higher, barkskin can give you +3 or more ac. The class ability enhance potion makes it your level. Amplify elixer extends it and allchemical allocation does not use up the potion. You just buy it once.
There is a good chance the barbarian is chaotic in alignment. Magic circle against chaos would work. That is +2 more in ac as a deflection bonus. Then, there is the shield spell if you can afford to cast the extract before combat.
The money down is to buy potions.
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So with cats grace, your base ac is 22. With the barkskin an ac of 26, if your opponent is chaotic with magic circle versus chaos an ac of 28. Wih the mutagen and ac of 30. Using a shield extract, an ac of 34. Maybe using an extract of reduce person and ac of 35. Reduce person does not hurt your bombs and you can burn the barbarian away with your bombs. provided you have fast bombs that is.
Combat Expertise is also an option; you have the intelligence, and mutagen can help shore up the attack penalty.