Since I have had shoulder surgery once and don't want another I don't want one kicking like a mule also gun weight is an issue as well for hunting up and down the eastern hills.
I started to fill in a bunch of options that I like. Then I read this part of your first post.
Based on that I can heartily recommend the .38-55 loaded with a moderately heavy cast bullet of around 260 to 280gns loafing along at around 1400 fps and shot from a lever gun. Or possibly something like and single shot like a TC Encore or H&R Handi Rifle.
The rainbow like trajectory you get from such a load isn't going to run flat. So knowing the distance and working to keep the shots closer in will be very important.
I personally shoot the .38-55 in that sort of loading from a single shot Remington rolling block rifle. And I've shot the same round from an 1886 lever. Both platforms give a light to moderate push. And it's noticeably more of a push than the Winchester 94 shooting the .30-30 I also really like.
Back that up with a good soft recoil pad and I'd be willing to bet that if the doc OK's you for any sort of shooting other than .22 that you would be fine with the big long .38-55.
I don't hunt myself but folks that do seem to think that anything that heavy running at around that speed is fully capable of traveling through a deer from stem to stern.
I see that a large number have voted for a .45-70. The times I've shot even the big and heavy Sharps with 45-70 loads I've only confirmed that it's not a round I'd want to shoot a lot of over the course of a day. And I'm not the only one that has nodded in agreement when I've voiced this. I've even converted a 45-70 owner to the idea by letting him shoot my Remington. And I don't even what to know what shooting them from a lighter lever gun is like.
Of course you can always load down. But by the time you get them down to being as soft as my .38-55 you'd be able to toss a rock with your good arm out further.
The other big vote getter is the .44Mag. And the primary way of delivering such rounds would more than likely be a Marlin, Winchester or Rossi lever gun. I'm here to tell you that full house .44's from these guns hit the shoulder at least as hard as a .30-30 from a 94. If you can tolerate that then fine. But be warned.
Another option is to shoot using black powder loads in one of the cartridges originally designed for black powder. Shooting the black will reduce the "CRACK!" and make it more of a "THUMP!". If you've shot any black powder guns you'll know what I mean by this. It changes the recoil from more of a slap to more of a sudden push. That might well prove easier on your shoulder as well. Especially combined with a good shoulder pad.