Spencer,
You provide an interesting question here. I don't trap so I don't get that close to an animal. However, I do hunt fox in the wintertime and I hunt those critters with a 22 mag rifle. Now, what you have to keep in mind is that I'm usually shootin about 50 yards at 'em. My maximum shot is about 100 yards so the 22 magnum with a 40 grain hollow point bullet does a good job on the red fox without pelt damage.
Close up though, (and I'm talkin 15 feet or s0, maybe 25 feet) I would think the 22 mag would still be OK,...but you may get some pelt damage with the magnum. However, on fox at that range,(25 feet or less) the 22LR should have no problem on 'em. Now, on a Coyote, I know a 22 Mag would be no problem at close range. Head shot or lung. A 22LR at that range with a head shot or lung should still do you alright. Remember, your not going to be that far away.
I think I would be looking as something like the Smith & Wesson revolver 22LR that has the 10 round cylinder. At least, here in Pa, you can't use a semi-auto handgun for anything except target, so a revolver that holds 10 rounds would be my personal choice. You know, just in case you trap that big 45 lb alpha male coyote! Maybe 4 22LR will do it.....but I always like X-tra's just for security!!!!
I think if you get into the larger calibers, your going to be looking at too much pelt damage. Even if your just shootin lead bullets. Lead will just give you a big hole in and a big hole out....but you really don't want any holes in those pelts anyway.
MHO
Good luck and I hope you have a successful season!
Dave