IF what you are describing is actually happening, I wouldn't be shooting the rifle. IF it is actually happening there has to be a structural issue with either the barrel itself (possible interior metal voids?) or with the barrel to action connection.
Either way, it does not sound safe and it is not normal barrel oscillation that a new composite stock will cure. If it were mine I'd get it back to Remington to sort it out before someone gets hurt.
On the flip side, how many days till April 1?