You would win because the U.S marine battalion has advanced weapon and better war strategies that those used by Augustus in the Roman Empire. Additionally, you already know where the Roman empire army errored in and thus you would seek to capitalise on these errors to win the war.
Rome would win. Everybody forgets that the ammunition the marine battalion would carry would run out. Plus a battalion is what, 500-1000 troops. The Romans had thousands upon thousands of troops. With the training of the time by the centurions and the lack of training by marines with those old weapons, I think the Romans would end up winning.