While I don't have a residential refer in my Montana, I have read about them Montana Owners Club forum. There is a disconnect switch for the inverter separate from the one in the convenience center. This one is located in the front compartment above the batteries. You wrote the refer has power when connected to shore power (which shows the builtin auto transfer switch is working)but doesn't when solely on 12V power. Assuming you found the on/off switch on the inverter itself, you should check the disconnect switch.