After the initial test, I set about mounting everything more permanently. I used some rivet nuts that are made for composite materials.
https://i.imgur.com/oig1r6h.jpg
They expand like a drywall plug, but are much heavier duty and really bite into the backside, as well as spread the load over a larger area. You can see the difference between a regular rivet nut for metal and the composite one. Lots of people use the metal nuts in fiberglass, but they have been known to strip and start turning if pushed too far. Of course there was a hitch (isn't there always a hitch?). My rivet nut tool mandrel was too short to grab the threads in the longer nut. I scratched around in my scrap hardware, cut a piece of tubing and came up with an adapter. It worked great.