I won't go through the exterior metal. If you have an extra panel that is not the exterior sheet metal you could attach to that. Make a bracket that extends over, with all sorts of ways to connect it.
Without "cutting holes in the carpet". I've made a mounting board out of scrap wood. Use Tnuts and you can screw the amp to the board, unscrew it. Then use t-nuts under the carpet and screw them in through the board. (Then mount your amp over the mounted board). If you drill the Tnut holes larger they will fit and be secured by the teeth. When it's all done you can take it off, and you won't leave any visible damage when it's removed.
(* cut a board to the size of your amp,
* Drill the mounting holes for the amp and install T nuts for them to screw into to mount the amp to the board, then unscrew the amp,
* Then drill holes around the middle of the board, put t-nuts under the carpet (2 is plenty), and (with the right length bolts, screw through to secure the board to the carpet with the T-nuts,
* Then install the amp to the secured board).