Problem with using a 22 rimfire on anything larger than small dogs, is it is illegal and unethical as well. I am sure a great many Inuits and their kin have shot game with diminutive cartridges. Doesn't make it the thing to do. I doubt seriously that a 22 rimfire lung shot 3000 pund moose is going to get much more than sick. Buffalo Bill Cody was the first to run herds of buffalo with horses using Colt .36 Navy cap and ball revolvers to kill them with. Bill, quit the praticeafter a short time, saying that a revolver didn't have enough killing power for the job. He went back to his 50-70 to feed the rail crews. Imagine if you wuld a 38 special at muzzles length not being enough for the job.
There are lots of legal chamberings that don't get the job done either, I have in my life of hunting killed 2 deer that were wounded by someone else, a friend took shi first elk, a 400 pound spike, had a broken rear leg, shot by someone else.............Please have some feelings and ethics about the gam yu take and how you take it...........(please don't read into that that last sentence that I think you an unethical hunter either)
Legal in a few states is the 223 Remington. the Handi doesn't have a fast enough twist to stabilize the heavier bullets needed for the job. Lots of people have killed deer with 22 centerfires, I will leave the question of that up to them....I don't.
I am a woods hunter. I see a deer for 15 seconds at max, and have a window of less than 3 to decide, and shoot. If you are close range hunting with a rimfire, this is exactly what kind of hunting you can expect.
I would suggest that you get a Handi in 30-30 or 44 Mag. Both are mild recoiling. GET A GOOD TRIGGER JOB done on it. If you don't already then begin handloading. that would not be a big jump, the 300 Whisper is basically a handloaded only proposition. Corbon I believe makes a few factory rounds for it.
The 30-30 with proper loads can perform in the same range of velocity as the 300 bullet, weight to bullet weight, many make light plinking loads of 900 fps and less and with a well tuned rifle and well tuned loads and carefully chosen bullets, it is easily capable of being a 200 yard deer rifle. Not bad for what is considered these days as a beginners gun.
I would however stay the hell away from shooting deer with a 22 rimfire unless you enjoy being fined, jailed and lightened of your property.