You assured the inverter is providing an output and tested every outlet in the camper, correct? As in, plug in a lamp into the outlet on the inverter, lamp lights, plug cable going who-knows-where into that outlet you just tested and then plug the lamp into EVERY outlet you find in the camper to see if any lights the lamp. Don't test one inverter outlet and then plug the who-knows-where cable into the other. When testing the camper duplex outlets plug the lamp in both outlets. It may be wired to where one is inverter and the other shore power.
Or maybe it just goes nowhere. You'll have to track it down
On Edit: Make sure you check the outlets behind the TV's