I would recommend one of the 6.5 rounds!
6.5x55 Swede is plenty for absolutely anything in that distance. Lapua and Norma make very fine ammo and brass. The brass can also be made from 30-06, 270Win, 280 Rem, 6mm Rem, 7 Mauser, 8 Mauser, & a few that aren't as popular worldwide!
6 and 6.5 Creedmoor has become popular, so brass has become readily available.
6 and 6.5 BR, 6 and 6.5 PPC, & 6 and 6.5 TCU have semi-wildcat status, but all will do what you are asking with great efficiency.
260 Rem and 7-08 have fallen into the "Has Been" category, good quality brass may become a problem, but you could make brass easily from Match 308 or Lapua's 308 Palma Match (Small Primer).
However you go, you should decide which of the bullets you want to shoot most and get the rifle with a twist to match that bullet. The 139/140 grain bullets use a very different twist than the 87/90 grain bullets! (don't really consider the 160 grain bullets, unless you are thinking of Rhino or quarrying rock!)
You may want to think about, how you plan on hunting. Either a light to medium stalking rifle, or a heavier rifle to use from a blind or bench. An 8 to 10 pound rifle is very heavy to carry through the hills stalking. A 6.5 mountain rifle beats the snot out of you for sustained varmint hunting!
There is no such thing as one rifle to do everything well! nor is there one cartridge. You will have to compromise somehow.
Ivan
(My compromise was to have several different rifles, handguns, and shotguns! I run from 17 cal pellet pistol to 450 Nitro Express double riffle, just to cover as many bases as I can!)