Recoil
Longboard--All else being equal (which of course is almost never the case!) the .243Win will kick far less than a .308Win or a .30-'06. The latter 2 have heavier bullets, and larger powder charges, which Ike Newton guarantees us will result in a heavier recoil.
Now, recoil is subjective, and also a heavier rifle will resist recoiling more than a lighter rifle. Also there is the consideration that an autoloader will sop up some of the recoil in operating its moving parts. And some rifles will fit you better than others--an ill-fitting stock makes the recoil feel worse.
All that said, the .243 is a traditional cartridge with which to start training smaller, more slightly built rifle shooters, and generally it works well for that purpose. My #1 Son started with a .243 when he was 11; it worked well for him. (He would NOT shoot my .30-'06 after one try.) It would probably be a much better bet for your torn rotator than either of the others.
There are now recoil pads available which help moderate recoil even further--Sorbothane (the gel shoe insert maker) has a new one that is supposed to be particularly good. If a factory stock is uncomfortable for you to shoot I'd certainly try adding a recoil pad as a next step.
BTW--Have you had someone take care of yr shoulder? You shouldn't just "tough it out;" physical therapists nowadays can do wonders, and if PT doesn't help, there is amazing new surgery. I had a rotator cuff injury myself, the PT was great, and the surgery consisted of 3 punch-holes the size of a pencil. Took a year to get over, though, and You Must Do the PT. I kept a daily log of exercises done, which amazed both the physical therapist and the doctor, and which got me better service from both I believe.