I don't like .308 recoil either and the cartridge doesn't punch paper any better at 200-500 yards than any number of other lower recoil cartridges.
There are a lot of other cartridges that will do a great job of punching holes in paper at that distance.
Last year, I researched the hell out of the "best low recoil medium range cartridge" question. I kept trying to get away from .243Win, because it's so damn popular and I like to dislike things that are popular and get something weird and interesting. Things like ammo availability and price for it were non-issues, because I handload.
Trouble is, every road kept leading back to the .243Win. As much as I didn't want to like it, it keeps being the answer to my questions. Plenty of other esoteric cartridges are out there, but the perceived benefits don't over come the shortcomings, like brass availability, or price of dies, etc.
Finally, I compromised and went with .243 Ackley Improved. I get 100-200 fps over the Win and still get my "weird" cartridge.
Still, the answer to the question is, .243Win. If you handload, the AI is great too. If not, the Win will do everything you need without punishing recoil, up to and including being a fine elk gun.
Still happily answering to the call-sign Peetza.
You do not HAVE a soul. You ARE a soul. You HAVE a body.
He is no fool who gives what he can not keep to gain what he can not lose.
-Jim Eliott, paraphrasing Philip Henry.