Our 5th wheel has a dealer added 1500 watt pure sine wave inverter and transfer switch to run the residential fridge when no shore power is available. There's about a 30 second delay built into the system before it will switch over from shore to inverter power when shore power is lost/disconnected. Of course, the batteries must be charged for the inverter to power the fridge, and the battery disconnect switch (the salesman switch) must be turned to the ON position.
If the switch is lighted on the inverter, that suggests that it's getting 12VDC power. If you're comfortable using a voltmeter, start at the 120VAC output terminals of the inverter and start chasing the circuit through the transfer switch all the way to the fridge and see where you lose power. If not, time to either take it to the selling dealer (since it's under warranty, right?) or call in the mobile repair service.
Rusty