If it was me, I wouldn't care if the fasteners I used to hold it were visible, and I'd use like an Dzus allen, or regular head 1/4 turn of some kind, spaced all in a line all anal like w/ a thin foam rubber strip in between the board and frame to act as a cushion for anti-rattle. If I just had to hide them, then I'd countersink them and use a plug on top. On top, and through the carpet just depends on what carpet you are using,.........I assume that you will use a "no back" of some kind since the area in question has to come up. It is very puncture friendly. Velcro would be my second choice, but the problem has always been the velcro itself holds better than the adhesive backing that holds the velcro in the first damn place. The cushion from the velcro will also serve as a sound dampener to eliminate any harmonic chatter that may happen. If you just had to be all stealth-like and sh it, you could epoxy a magnet to your substrate, so it shouldn't let go of it's backing before the electro bond gives up when removing,..I just think you'll get a chatter from the hard surfaces . If you absolutely don't like the visible option, Go w/ one of your hidden options first,....it's un-doable if it wont work, and will not leave any sign of it being there should you have to improvise.
The things is,...I have to sit and look at it for about 4-6 hours before I come up w/ a finished plan, and most of the time "plan A" gets modified when it doesn't quite work for me. Just like fastening the brake line to my own rear end project,....hose clamps gave way to riveted SS line clamps on the housing after I got dogged by the consensus for cheesing out.
Love the finished I.R.S stuff,.....I'm