Wanting to buy a rifle, but have a bad shoulder. Would be for hunting/fun.

My shoulder was basically rebuilt 3 years ago, and I still have a permanent restriction on it. It gets sore fairly easily. For a reference point, in December I fired 3 or 4 1600 fps rifled slugs from my 20 gauge and my shoulder hurt/was sore for the next 6 days. I will say it was an old crackbarrel wooden stock, manufactured before 1958, and with no hint of padding. Still shouldn't have been that bad.

Seems like my options are .243, .300 blackout, maybe a 5.56 with no margin for error, maybe 7mm? What are some other options which might work for me? Just don't give a suggestion like my FiL. He insists a .22 magnum is a deer gun, while only using it for headshots on deer that walk into the yard around his cabin.

Price is also a concern, the cheapest way to get a rifle I could hunt with would be a Mosin Nagant. But I'm guessing it would beat up my shoulder way more than I would want to deal with.

I would really like an AR. Obviously plinking would be the most fun with it, but I could get a .300 blackout upper at a later date to swap in and out. Can't really afford one right now though, unless ya'll know of some decent deals on parts to build one.