My dad use to trap in the 60s and 70s, he carried a Ithaca M-49. He used the cheapest 22 ammo he could find, in the late 70s I remember seeing Thunderbolt and Wildcat boxes in the garage. I guess from a few yards, it didn't really matter how acute the cheap ammo was.
I'm not a trapper but did trap some rabbit's to eat while living in Alaska. The gun to carry is whatever gun you like! Love my Marlin 22 mag but for that I'd prefer my old Winchester mod 62a. Better yet would be my High Standarg HD Military or my mod 16 S&W in 32 Long. Handgun's make time at the trap almost 100% hands free!
I used to run a trapline with a four wheeler. Had a ruger m77v in .22-250 in really stout protective case mounted on it for critters I saw in distance. Carried a Beretta minx .22 short for dispatching raccoons and such. Carried a ruger stainless 5.5" mk 2 target for in between and skunks. Think that smith wesson 422 another poster mentioned would have been about ideal.
Carry a Winchester 70 22-250 with a 3-9 Nikon on the snowmobile or 4 wheeler in my pocket a S&W model 63 going back to the late 70's now a S&W 317 with a 2" barrel it weights only a couple ounces and loaded with sub-sonics it will put down a coyote and put a grouse in the pot....
Bought me a Ruger Wrangler 22lr. They come with cerakote from the factory and I have been really happy with it. Use Cci Quiet22 ammo and have not had any problems dispatching 'yote's. Used to carry a Ruger 22/45 with a can. Still do occasionally when setting coon traps around the deer feeders to keep the noise down.