The 300 Winchester Magnum is the way to go, unless, for some reason you want to avoid that particular chambering.
I personally favor the 308 Norma Magnum, but it is a ballistic twin to the 300 Win, so it's a moot point.
My reasons are multiple.....I have been shooting he 308 NM since 1965, I like the slightly longer neck of the Norma design.
I also have several rifles so chambered. I have enough components [Brass] to last this plus another lifetime, lol.
The 300 WM, and the 308 NM are both very accurate chamberings, and have enough power to take any game in N.A.
plus most of the world. OP, Of the two you mentioned, the 300 Win Mag gets my nod. D.