I was going to suggest you look into the 7MM-08 also. It is on par with the .270 and .308 also. The 7MM-08' s advantage is that it is a short action caliber and does have less recoil that the others. It also has more retained energy down field than the .308.

I'd take the Remington Mountain Rifle DM in 30-06

I use a Remington Mountain Rifle in 270 cal with 150 grain nosler partitions to hunt elk on the eastside of Washington.

For my 2 cents... I think the 308 or 358 are rerally good choices. I have the medel 7 in 308 and the blr in 308 and 358, none of them have recoil that should bother you, and the rifles seem to fit me well and i have short arms for a guy my size. either of the calibers will do the job nicely with either factory or hand loads, ( the 358 factory is a 200 gr silvertip Vapo, from win). The 358 BLR is my newest and is becoming my go to gun. Groups well and has enough energy to do what is needed, it seems to put animals down better than the paper numbers would have you believe, i do handload for it though, but the first hunt was with factory. And all the above rifles are light, just a little over 6 pounds, but none seem to have objectionable recoil. Good luck with what ever you choose.

The .308 Win. be the most logical caliber for you, as suggested by Vapodog. Good luck.

Remington 7600, light maneuverable, chambered in most standard calibers.