Strosfann, I see you are from Tomball, hometown of the greatest tenor to ever sing Christian music, David Phelps. He's awesome!
Your son just needs more range time with the .308. Lightweight rifles require much greater attention to form and trigger pull than heavier rifles. It takes more force to move a heavier rifle, and much less force to move a lightweight. Even your heartbeat exaggerates the movement of a lightweight rifle vs. heavy rifle. Your son will need to learn better shooting technique to be successful with the .308. It will come, and in doing so will make him a much better marksman. So don't give up. Lightweight rifles are harder to shoot well, but the journey is worth it.
Keith