Nothing fancy is the key - 308 is the better option.
Given a bit of 'fancy' in bullet choices, I'd take a 223 over 308 in the none-to-medium wind conditions, just for the reduced recoil. In a low-wind condition at 350yds, the ballistic advantage of a heavier 308 bullet is minimal (much less than an inch is my gut feel - have to do some calcs tomorrow).
Given a bit of 'fancy' in bullet choices, I'd take a 223 over 308 in the none-to-medium wind conditions, just for the reduced recoil. In a low-wind condition at 350yds, the ballistic advantage of a heavier 308 bullet is minimal (much less than an inch is my gut feel - have to do some calcs tomorrow).