If you want RF sheilding to work, it needs to be electrically continuous around the whole encloseure. Separate hunks of sound barrier with Al layers in them just turn into independant antenna's, even if you ground them together (not easy to get a good grip on the foil layer anyways). That's why normal tinbox cases have all those funny little tangs/dimples around the outside, to connect the sides-back-covers etc. together at regular close intervals.
Lots of plastic cases are internally spray coated with conductive paints. I have done a wee bit of preliminary looking into these paints, and they appear to be nickle-flake, or maybe Al in budget/bulk versions, with silver-flake available ($$$$$) for critical high grade work. I think you could get fairly good results on wood with something like that, and it would be practical and cheap if the right paint can be found. It must exist.
I'll post back further when I get around to searching it out. In the meantime MG Chemicals does have a product, albeit expensive.
MG Chemicals: Super Shield Conductive Coating