But the shotgun won't get you the distances you are talking about. If you are really going for beyond 500 yards, I'd say 6.5CM has become America's caliber (including SOF), but 308 will do.
Ruger Precision is worth looking at. SIG Cross is a bit lighter.Special Operations Forces appear to be increasing their focus on the 6.5 CM as the best way to take advantage of advances in ammunition and weapons technology to improve intermediate range sniper lethality, reliability and performance — when suppressed — at distances from 50 to 1,500 meters.
If you are looking for a non-semiauto fighting rifle, old school WW1/WW2 rifles are still around.