Sounds like the same "sandwich" or laminated construction used on many RVs for their roof and sidewalls? Just like cardboard, it works great until it gets wet. Then, it can be pretty difficult to repair.
That rubber membrane may have unintentionally prevented water from draining? Obviously a very important part of this repair is going to include finding and fixing the reason that allowed water in to start with. Is the water tank or any other plumbing under the bed?
If this problem is between the rig's frame rails, your idea of adding stringers between them and building a new floor on those is a good one. Depending on the damage and size of the affected area involved, overlaying with 1/2"-3/4" plywood might be an option as well?
1997 37' HR Endeavor, 275hp Cat, Freightliner
03 CR-V Blue Ox, Ready Brake