[RMcNeal]

Tree Top Turkeys, it sounds like you have a residential refrigerator like mine. If so, the inverter should have a separate connection to the battery(ies) with its own cut off. Mine has a round turn style cut off in the battery compartment on the back wall. From there, the wires go directly to the inverter. If you leave that on while towing, then your fridge will run on battery power. If you leave it on while on shore power, the battery will power the fridge in the event of the electric going out. There should be an auto transfer switch after the inverter which uses shore power, when available, then battery power when shore power is not available.
The red key in the wet bay has nothing to do with this circuit or the residential fridge. However, if you turn it off, where the key can be removed, then your converter will not charge your batteries on shore power. I leave mine on all the time and I installed an extra battery disconnect on the negative post of my battery box to completely disconnect power when in storage.