I would think a 45# recurve, longbow, or short longbow(oxymoron) would fit the bill. You wouldn't have to pull through the "violent" cycle of the compound, which feels like it would want to rip your shoulders out of their sockets! A 45# recurve with a razor sharp broad head will pass right through a whitetail.
I used to frequent a traditional archery website and almost everyone over 50 was experiencing the same types of shoulder situations as you. It's unbelievable how much good can be done for shoulder health by doing just back side pull excersises. We don't even need weights. Just get a set of bands, the same kind they use at the physical therapy place, and do face pulls, chest pulls, reverse fly's, etc... Turns out all the wide grip benches, deep dips, behind the head presses do a pretty good number on your shoulders. Lots of great info out there on the web about shoulder health, just have to navigate through the bull****. Good luck with your rehab and your future bow hunts!