Gun nuts --like me!-- have a knee-jerk reaction to follow the bigger-is-better mentality. Pragmatically, you want to follow a "Goldilocks" principle: not too big, not too small. I never would have considered using my .22 rifle for shooting the coyotes who raided my chickens (it would have risked wounding rather than killing), but nor would I have used my .308 for shooting the squirrels raiding the vegetable garden (it would have been ridiculous overkill and torn craters in the garden).
Would the 12ga be more forgiving of bad aim or a nervous flinch? Yes. Is a 20ga adequate, if you do your research and then aim properly with a steady hand? Yes.
Take into consideration your skill, your preparation/education on technique, and what you think your mindset will be when the time comes and whether you'll be steady enough to make the shot you need to make. If you're winging it on aim/anatomy, or think you'll have the jitters (understandable!) and not be able to concentrate, you might want the degree of overkill a 12ga provides. If you think you'll be steady and focussed, and you are confident in your knowledge of skull anatomy and aim, then the 20ga should be just fine.
Also consider the basic ergonomics of the gun. Can you point it where it needs to be pointed without fumbling it? A full-size 12ga is a terribly unweildy thing to put in your mouth, which makes it more likely for you to slip and blow off your face. Not good. If the 20ga allows you more certainty of being able to aim accurately, then the 20ga will be the better choice.
It's up to you, of course, and bigger-is-better is not without its merits, but take all the factors into consideration.
