Any 7 mag with a 180 VLD should work fine. Any .308 mag with most 180-210 bullets should work fine too.
Some smaller calibers would work with good shot placement but it's always nice to have a little extra horsepower when shot placement isn't perfect.
A custom rifle
isn't "necessary" but an accurate rifle is. You can only know how your rifle/ammo shoots at distance by actually shooting at that distance. A rifle that shoots 1.5" @ 100 yards will almost always shoot groups more than 8X larger or more than 12" @ 800 yards despite what the math tells you. (1.5" X 8 = 12")
Barnes bullets probably need 2,000 ft/sec on impact to expand reliably. What would the impact velocity be from your 165gr Barnes at 800 yds?