The 7MM Rem Mag will do fine for what you want to hunt.
Mine is a Model 70, and at this point, everything on it is still factory. At 1,000 yards, I get a vertical spread of 7-8". I'm still having problems with reading the wind and my horizontal spreads go from 8-24" depending on how bad I screw up reading the mirage.
I hunt coyotes, not much else. Their small kill zone so far has limited me to under 800 yards.
With some decent handloads you should easily be able to reach your goals. Try some good ( Norma or Lapua ) brass, and some SMK's. I like shooting the Hornady 162 A-Max and the Berger 180 VLD's too, but so far I'm not sure how they will work out on game bigger than Jackrabbits.
So far the A-Max make holes WAY too big to be acceptable, and the Berger's go the other direction, the bullets pass through with out opening at all.