I'm wondering if the are partly to prevent feedback (by muting the vibrations on at least one face of the body)?
I'm willing to venture a guess that the pampers attached to the back is to help the body from resonating and feeding back, sounding like a freight train horn.
I first encountered these gizmos on the late-60s Vox Apollo IV a bandmate in junior highschool had 40-odd years ago, and I gotta admit, never once did it occur to me that the pad was there to impart some sort of sonic effect. But if I'd never seen what was underneath the "barstool cover" I guess that would be a plausible explanation: a resonance damping device. I applaud your creative thinking.
However, at least on that Vox bass what's underneath the foam pad is a plastic lid to allow access to the electronics compartment, just a thin unsupported piece of polystyrene that would most certainly get scratched up (if not caved in) by the enormous belt buckles that were popular at the time. So I'm fairly certain the original intent was simply to prevent the bass from getting dinged.

