@BigNate, you brought up something that reinforced the cartridges I listed... Reduced capacity in the magazine, compared to .308 Win capacity. If you build off a .308 Win parent case, the capacity stays exactly the same... It's the same case, just necked down...
And that weirdo 6.5 creedmoor bastard is still 20 rounds in the 20-rd mag, too.
.260 Rem is the TICKET, man. Those .264" projectiles are magic, but the real question is... is that something that you want to reload? Finding ammo is NOT really an option, because all the commercial offerings were for hunting, in a bolt gun. Reloading it is easy, if you're already reloading. After you size it down, you'll have to turn the neck diameter down, but I can tell you the exact tool that you'll need, and where to walk in and buy it locally. KM Precision neck turning tool from Bruno's.
If you don't want to reload for it, then 6.5 Creedmoor is the obvious choice - TONS of good gas-gun ammo produced by manufacturers.