If you want to consistantly range to 800 yards and especially deer or game you will not get one for "$350 or less" I would advise you to save a few more bucks and watch for a used Leica 1200. You can get a used 1200 CRF for about $450 and it will be money well spent. If you buy new for $350 or less you will be wanting something better real soon.
Bushnell 1300 G-Force will fall in that area. Mine will barely meet the criteria you have put forward, on small animals anyway. Antelope, deer and larger no problem.
However, the Leica would be hard to argue if you want/can spend the extra for it.