If you want ta go with what has a good chance of making a humane kill, .177 is out.
25 FPE is a minimum, probably want at least 30.
That means a decent performing .22, but probably .25.
If you're gonna be consistently huntin coons, I'd go with .25.
If it'll just be for the occasional target of opportunity and you prefer it, a good .22 will do.
