I think a .308 Win. or a .30-06 would be just about perfect. A rut crazed elk is a tough customer. As are Mountain Goats. I've owned a .300 Win Mag that was 7lbs ready to go, and the recoil with stout loads was noticeable, but fine for me.
If the ones I posted are too much and you need very light recoil I would look at a .257 Roberts, .25-06 or 6.5x55 SE.