Proud Dad wrote:
.. My theory (that is to say my hope b/c it would be a simple solution) is that there needs to be good DC power in order for the refrigerator controller to work correctly even with LP gas as the energy source, and that when I replace the batteries, the fridge will work just fine.
Fridge controls work on 12V DC. They control the 110V AC circuit AND the propane side.
Your converter is providing the 12V when you are connected to shore power, but it will not charge a battery that is toast!
Batteries should solve your problem.