how about a sheet of plywood glued to each side. Cut about 2" off each
side so the edge of the foam can act as a bumper Then use threaded rod
to through bolt your cleats. The bolts would also keep everything
together if the glue failed.

Thardware stores sell adhesives for foam. I stuck foam insulation on the
concrete basement walls, then sheets of drywall over top with the stuff,
back in 1980. Still sticking to the walls. Applied with a caulking gun.