For the carpet, I would probably try initially just cutting it carefully to shape and seeing if it stays in place on its own without any adhesive. My guess is the needed notches and jags in the outline would do a pretty good job of keeping it in position adequately.
If some sort of adhesive is needed, I think contact adhesive or similar is what I would try rather than double-stick tape, at least the foam double-stick mounting tape. The foam tape would cause ridges in the "floor" of the cabinet due to its thickness.