After some thought, I think you would be happiest with carpeting the whole thing. The carpet will tend to dampen the box reasonance. Shoot with that box I would just carpet every surface of the interior. I did this with a D-50 (Dodge) pickup I had several years ago. I built a box, hooked up a signal generator to find any imperfections, and sealed them. Then I carpeted the box and any exposed metal or surfaces which might tend to vibrate. It sounded pretty good. I had 150 watt 12" subwoofer that played through the seat. 70 watt midrange door mounts, and 30 watt tweeter surface mounts. It sounded good with NO tinny resonance sounds. Two songs I use to test this are Back in the high life, Steve Winwood- Has good high and lows and strong bass in spots. The other is shoot to thrill, AC/DC the bass drum solo in the middle will shake everything that might be loose so you can find it and tie it down.
My electrical engineer friend who helped me with my pickup said "remember, loud is not good unless it is clean".