I would suggest you check with your local city.....in my experience MOST cities make no difference between your 50bmg and the red ryder. Yea stupid I know....and most deputies here if called will do everything they can to not bring you to jail....but that is different from place to place and notice I said MOST. It really depends on a great many factors.
I do agree on a 22 Pellets are very different from a bullet. And a 22 pellet is on a good day going 800-ish FPS AT THE MUZZLE with a 14-ish grain pellet with the BC of a brick. Even a 22 long, not long rifle is flinging a 40 grain bullet at 700-ish fps....that is a LOT more mass and a lot more energy on target.
One year I used a 20 ga shotgun but some neighbors complained.
I killed my first chrono doing testing on pellet speeds at target, Down load a copy of chairgun, from Hawk scopes....if your pellets BS is not in there do some looking on air gun sights they are out there. Point is a pellet slows down quick, and the flight of a pellet really goes to hell if you push them supersonic, worse then a rimfire 22....ever wonder why target....good target 22, like RWS, Lapua Eley...are sub sonic....because pushing them fast will hurt how accurate they are as they slow back down....you don't need it to put holes in paper, you are not causing damage to a living target.
We all know foot pounds when a bullet hits something is all math with weight and speed being what really matters.....with pellets you have light weight and low speed, so not a lot of energy.....you need to practice and hit the spot on the critter that will kill it.
This gets me on my high horse faster then anything else....and it is not directed at the person I quoted....I am just rambling.
Know your rifle, know your target, know what you need to hit to put the animal down clean and ethical manner. Generally we are talking a quarter sized target area....figure out where you can hit that sized target and that is your range with that air rifle.
Now as you move up to larger air rifles, 25,30, your margin of error gets larger....
I really don't hunt anymore, and now that most of my animals are gone I don't really pest anymore, but I can tell you I took a shot on a mouse, not a rat but a field mouse with a 17hmr, and ended up blowing the back half off of him....he was still alive, makes me feel real bad, no animal deserves to die in pain, no matter what you say he did, it is just an animal doing animal things...sure rid yourself of pests, but do it in a way that does not cause suffering.
Sorry for the preaching post.
Man I am tired.