[RMcNeal]

Dobie Dogs, If you have 120 VAC at the outlet on the inverter and the fridge doesn't get power, then the problem is most likely the transfer switch. If you can read 120 VAC on the inverter's outlet, then the GFCI is not tripped. The only thing in the circuit is the transfer switch which is apparently not flipping over when shore power is removed.
Incidentally, if the fridge is good and cold, it may not need to run the compressor for a while, in which case the inverter may not show any draw. Easiest way to test if it has power is to open the door and see if the light comes on. These Samsung fridges are very efficient and use a minimal amount of amperage. I was amazed with ours.