Medium game to me, is mainly black bear, deer, and pronghorn. 6 ARC or 6.5 Grendel would probably get the nod in an AR-15 platform. The both have plenty of power and range to handle the three game animals above without much fuss. If you want a little more exotic then you could look at 6.8 SPC II or .358 Yeti.
If you're going to step up to AR-10 while I like the .308, I'd probably go with 6mm or 6.5 Creedmoor. I do prefer 20" barrels even in the .308, but the AR-10s are a bit heavier. If you're willing a carbon fiber wrapped barrel allows you to use longer barrels without increasing weight.