I don't think that the OP is trying to take game at 800 yards but he is looking for a gun that will both shoot targets at 800 yards and shoot game at typical hunting distances. At least that's how I read it.
Trouble is that he's going to end up with a compromise for both goals unless he decides to focus on one goal and take what he can get for the other.
I'd like to ask the OP how often he'll be hunting versus long-range (or mid-range to some people here) target shooting. If the honest answer skews heavily toward one or the other, optimize your choice for that and take what you can get on the other.
I suspect that many people can get to the range more often than go hunting, which is too bad because the OP's budget lends itself more to a hunting rig versus a long-range target rig.
I'm a Tikka fan but Savage seems like the leader in best performance for the price right now so I'd look at a hunting rig with a Savage in .308 and a Vortex/Nikon/Leo scope. If you want to try to cover both bases you could move up to a 300 Win (the other cartridges are better choices if he reloaded) but that's not ideal for both either.
Best of luck,