You complicated the matter by including elk in your question - I believe most hunters would agree that elk are inately tougher critters than caribou. One reason is body weight, a really big bull elk is far bigger than a bull caribou, plus sometimes elk just seem to take a lot of killing.
I would suggest you could go down to the .257 range, my son has taken a lot of caribou with his .257 Weatherby and they drop dead. The .257 Roberts or .25 WSSM would do the job almost as well. Matter of fact I expect the .243/6mm with the Nosler 95 grain bullet would also kill caribou very cleanly.
If I was recoil sensitive I would consider a Win. Coyote in .25 WSSM. You could also consider the 7mm range of cartridges - hard to beat a 7-08 for a nice shooting cartridge. Again you could do the Coyote in 7mm WSM
, that would be a heck of a long range rifle and it would not recoil much. I find that the 7mm WSM kicks noticeably less than the .300 WSM
- due to lighter bullets mostly.
Living up where you do ammo availability would have to be considered - of the above the .243 would probably be the most available. What about the .308 Win? In a medium weight rifle the .308 does not recoil that bad, and it is a proven killer. Use 150 grain bonded bullets and the .308 is a killing machine out to 600 yards.
None of the above are good choices for elk tho - although lots of guys kill elk with lighter rifles the .300/.338 mags are the best choice.