Those panels can be fabricated as you suggested out of hardboard/Masonite. Fabric glue or Elmer's brushed thinly seems to work well to glue carpet to new panels.

A little work with some epoxy or high strength craft glue can secure those trim clip fasteners into a flaky original panel.

Finally, one of my favorite tricks is to replace the trim clips with small neodymium magnets glued in place. The local craft store carries them. I would only recommend using the nickel plated neodymium magnets, not the cheaper black/grey ferrite magnets. I reattached a few trim panels with magnets years ago and they stay in place perfectly until you need to remove them - then they just pull off.