Howa 1500 is a good action to build on. I wish they made them in LH. The Rem700 is fine as well and there is heaps of bits for both.
To kill shit proper @ 800m you will need something Magnum. The 300WinMag would be my pick of the two options.
A build is going to cost you a LOT more coin than $600.
Don't forget you are going to need good glass to shoot at range and you are going to need to drop at least 1K plus to do this. (Sightron SIII would be a good start).
Shooting at distance is an expensive hobby. Add to the glass department good bino's and a rangefinder too.
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