Matt, what's your favorite mounting clip? So far I've seen the wire type and the Chrysler sheet metal type. If I use foam padding, it would probably only be 1/4" or 3/8" thick, at least around the edges, and I'm thinking that it needs to be fairly firm. Seems like the clips might still show through thin foam.

Ultimate Guide to Baltic Birch Plywood: Why It’s Better, When to Use It «

This place seems to have a lot of foam types, thicknesses and grades of firmness.

https://www.foamorder.com/