[Carl n Susan]

I doubt you have a Transfer Switch problem as the refer works fine with shore power. The problem occurs when shore power is removed. That implies the inverter is not sensing the loss of power and shifting to to battery power. I believe the refer is always plugged into the inverter. The inverter is the one that decides what power source to use. Obviously shore power is the preferred source. Lacking that, 12V is inverted to to 120V to power the outlet.

Take a hard look at the power source and output of the inverter.