The 7.62 is similar to the 30-30 in performance, either would work. The recoil would not be much lower than a 243 but if building a rifle the 6mm BR would be about ideal with 80-85 grain mono bullets, the same for a 6x45 or 6x47mm. Nothing wrong with the fast and faster 22s either if legal.

A sorbathane recoil shoulder pad and a good butt pad might be enough. Try these first before going to a muzzle brake. Off a good rest I can use either shoulder but the left eye dominance causes issues for me sometimes, I have seen both off set scope mounts and off set gun stocks made to overcome this. I am probably looking at shoulder surgery some point in the future. Meanwhile glucosamine and hemp oil seem to help and when really bad I have resorted to a cortisone injection before hunting season.

"When you disarm the people, you commence to offend them and show that you distrust them either through cowardice or lack of confidence, and both of these opinions generate hatred." Niccolo Machiavelli