I had Factory to some work done to my Montana that required slide removal. After its return on a trip I discovered same issue of Res. REFRIG. NOT powered when off shore power. FACTORY had hooked up 110v power leads in wrong order to the slide. This put the refrig 110v circuit with its Inverter on the TV circuit. I discovered the 110v TV worked fine without shore power.
I see your TV and Refrig. are in the same slide. IF YOU have had slide work done possibly same problem has occurred.
TROUBLE SHOOT, I checked my inverter and it SUPPLIED 110v when off shore power. But refrig. Was not powered. When on shore power with inverter OFF I switched of breakers one at a time to confirm what lost power discovering breaker marked refrig. Shut of TV and slide power breaker shut off Refrig. With inverter on breaker marked refrig. It was time to check wiring once I remembered they had to remove slide. Turns out. (3) 110v circuits had power leads connected incorrectly from markings on wires and the terminal block under the floor.
I certainly do not know your RVs history so some simple trouble shooting may fine your issue. The Inverter has to be wired to same 110v wires as circuit breaker and refrig. The 110V INVERTER POWER. If power lead goes to wrong outlets like TV then refrig. Won't work.
BEST LUCK
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