Not sure if your buttplate or recoil paid will come off, if so you may be able to access the inside from that direction. On my .30-06 Rem 760, I took a small plastic medicine bottle and filled it with lead shot, and taped it shut. In the stock hole that is there to give access to the bolt that connects receiver to buttshock, put a piece of cotton, then drop in the medicine bottle, than another piece of cotton. This does two things: (1) adds weight to stock; (2) give a piston effect that absorbs recoil when the medicine bottle slides back when firing. The cotton keeps the movement silent. I'm not sure if you stock has this same opening or not, but all wood stocks do.