Depends on the condition of your shoulder. I had mine rebuilt about 20 years ago, I feel it grinding around a bit, but I don't notice any issues from it with recoil (I live in a shotgun-only state for deer and the recoil from a 12 gauge slug gun is fairly substantial). And, is your shoulder a temporary condition? Do you want to make a major purchase based on something that's going to resolve in a year or two?
Anyways, I always hear about the Swede's taking moose with the 6.5 calibers (.260 rem would also fit here), so I suppose that would work for elk. I own a 7-08 and like it, though I bought it more as a deer/antelope gun (I see plenty of posts saying it will work for elk, though). Problem with some of the rarer calibers is ammo availability and price. Of everything mentioned so far, the .270 is going to be the most available.

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Some more objective guidance:

http://www.chuckhawks.com/recoil_table.htm