It sounds as if the self-adhesive was intended to be the primary method of fastening, and the staples were there to reinforce that.
Perhaps try this:
- Reverse the strip and use the self adhesive as intended;
- Rather than staples, reinforce with screws;
- Since screws don't old well in "pressed board" or particle board, inject Gorilla Glue into the pilot holes and onto the screw threads. Gorilla glue will bond to both the "wood" and the metal and keep it from vibrating out. I use the Gorilla glue tactic on every fastener I drive into the "wood sort of stuff" commonly used in RVs...from cabinet hinge screws to small lag screws. It works very well, and with a cordless impact (screw) driver, you can break them free to do repairs as needed.
Note that Gorilla Glue expands much like aerosol expanding foam insulation, so apply it sparingly. If it does ooze out in a cosmetically sensitive area, do NOT wipe it off while wet. It will make a mess. Let it dry then scrape off the excess.
I'm not sure this is possible, but staples are about the worst fastener you can use in virtually any application.
Good luck with the repair.
__________________
Jim & Renee
2020 Jayco Jay Feather X-213
previously 2014 Forest River/Rockwood HW 277
2006 Ram 1500 4WD Crew with Firestone Airbags
Every weekend boondocking in the National Forests or at
Lake Vallecito.