Just Marsh,
Here are some options:
1. run the fridge from the inverter, but disconnect the converter.
2. run the fridge on the "automatic" setting which allows it to choose propane when not connected to shore power.
Unless a special effort has been made with the charging system from the alternator, it is unlikely that the batteries will stay charged using the inverter to power the fridge. My fridge uses 325 watts when cooling and on an average day uses 5.7 KWH.
Allowing for inverter losses, 325 watts works out to a load of over 30 amps.
Unless you have quite a good solar installation running the fridge via the inverter may drain the "house" battery bank.