Do a Ruger stainless Scout in synthetic, very light and accurate. You can add the rail that has the rear sight and use a traditional scope. Choose your scope/rings wisely and it won't have to bad of a cheek weld.

Tikka makes a .308 model that's a spin off of the one they make for the Canadian Rangers.
https://www.recoilweb.com/tikka-t3x-arctic-131608.html