We leave it on "auto" when traveling. On my 2010 2551, the inverter does not operate the refer, so it is always on propane when traveling. When we get gas, my wife turns off the refer while I fill, I think this is important as the propane flame and spark igniter are only a few feet from the gas fill.
I am surprised that your refer runs on the inverter. You can verify this by switching the refer to manual AC while not plugged into shore power, on mine, the display goes to "A" indicating no AC available even with the inverter on.
Bob