The Winchester Ballistic Silver tip ammo is very good but it is designed for hunting, not match shooting. I have some of it and it is pretty accurate and I dropped a deer with one shot easy enough. For what you are describing I would get a box of the Federal 168-grain Gold Medal Match ammo. It is loaded with the Sierra Matchking BTHP bullet. I have bought a lot of this ammo and found it to be the most accurate factory loads I have tried, even more accurate than the Lupua Aficionado+. I shot a box of this ammo through my chronograph and there was very little variance in velocity for the 20 rounds.
I have a Savage 10FLP in .308 that is very accurate. If I were shooting against someone else shot for shot with factory ammo I would definitely go with the Federal Match King. If you and your friend are equal in skill level as shooters you should have the edge. At 1000 yards I would give the edge to the .308.