Done it in the past, before I foresook (sp?) dino-excrement stocks.
One I just filled the butt with the low-expanding big-gap filler & one I cut slices of blue closed-cell foam insulation & then spray-foamed the gaps.
Either will cut down on the noise factor, but if I remember correctly, the blue foam was better (quieter?) than the other.
If you're doing it on a Tupperware stock fore-end, I'd use epoxy stiffened by redi-rod in the fore-arm, in order to provide stiffness, especially if you are going to use a bipod/rest.
If you do that, though, you're going to have to (well, you really do not have to, if you don't mind an out-of-balance shovel handle) experiment with weighting/balancing the stock.