On my 335DS the fridge is wired through a transfer switch (small black box) that both plugs into the inverter and is hard wired to 120 volts. It looks like this.
If the fridge doesn't work on shore power OR battery (presuming the inverter is turned on) the failure has to be the transfer switch itself. I haven't looked to see is there is a plug in the fridge in my unit or the switch is hard wired to the refrigerator. You need to follow the two black wires from the switch. One will go to 120 VAC power and the other to the fridge.
The transfer switch (a relay) senses that there is shore power and switches the fridge automatically to show power and disconnects it from the inverter. The inverter will continue to run but with no load.