What are you looking for the character to do? Other than both using WIS as a casting stat I am not seeing any synergy between these classes. The most important thing in a gestalt game is to choose two classes with good synergy. The gold standard is to get full BAB, all good save, 6 skill points per level, 9th level casting and class abilities that work well together, and can be used together.
If the class abilities have really good synergy than missing the gold standard is often ok. Unchained Monk has really good synergy with the inquisitor so dropping down to 6th level casting does not cause a problem.
If you want both classes to have magic you might want to consider Hunter instead of alchemist. Both classes are WIS based spontaneous casters. You would be sacrificing full BAB to gain 2 6th level spell lists. Unchained Monk would get you flurry of bane which is insanely good. It also gets you both stalwart and evasion.
Yeah, I tend to think about gestalt very differently than others seem to… for me, gestalt is just getting to mix 2 things you’d like to play, as long as they aren’t mirroring each other. My favorite gestalt is a Kineticist and vanilla Summoner. All sorts of spells that complement my Air Elemental theme, and an Air Elemental partner for those times I need a linebacker to tackle a threat so I can do what I need doing.
If you have your martial set, and support is your gestalt, you need a class that can do their support without taking up your combat rounds. Paladin Warrior of the Holy Light is an idea… being able to give up spells because you already have spells is easier than playing that archetype without gestalt…
Btw, just wanted to add… my summoner gestalt has not a single summoning spell, not a single offense spell, not a single defense spell… if you have a solid primary, you can have a lot more leeway on what you do with your gestalt…
I would go with paladin/ranger. Maybe with an oath against undead for the paladin. Favored enemy (undead) and smite will make short work of vampires and their spawn.
Paladin/Ranger is going to be incredibly MAD. They will be limited to medium armor so will probably be forced into a DEX build. They will need both WIS and CHA for spells and only get 4th level casting.
There are lots of powerful gestalt combinations. Just so many. And people commenting here will have their personal preferences. But what do you like to do? Do you prefer half-casters? Are you wanting to be in melee more or less? How much do you want to sacrifice to specifically focus on killing vampires?
So far you've expressed interest in 4 classes, but I'm not familiar with the factotum nor sure which version you are using. I would suggest that you prioritize getting as many good saves as you can, and if you have any plans of attacking then you need to have full BAB in one of your picks. And it's a good idea to not mix requirements on mental stats. For example, you don't want to have abilities based on intelligence competing against your abilities based on wisdom. That said, some classes have no real mental requirements, so don't ignore those. Therefore, a vampire hunter/barbarian wouldn't be much of a caster, but otherwise would still be a good combo.
Vampire Hunter
* Pairs well enough with inquisitor.
* The herbalist alchemist is wisdom based, so should be a decent pairing.