When I served in the Military we shot the 308 to 1000 with peep sights. It’s do-able, but aren’t scopes awesome though. I love scoped 270 for this task. 110gr Barnes leaving the barrel at 32-3300FPS will DRT a whitetail for a joke. Less recoil then my 308 with a 150gr pill.Since you want to shoot targets @600 and game at 400, don't enjoy recoil, then the obvious choice is to build a long action 308 Win. Same bolt face.
The 308 Win was used extensively at 1000 yds in its run at match shooting.