Have not tried this, but did a little research and thinking about it. I believe the Dometic 120 volt heater element is 325 watts: http://www.amazon.com/Dometic-120V-R...productDetails
As such, I think a 400 watt inverter should work. Also, as the heater is a simple resistive load, a "pure sine wave" inverter should not be necessary. A 400 watt inverter supplying 325 watts will likely draw ~30 amps. As the refrigerator will be "on" only when required for cooling (assuming you pre-cool on shore power or on propane) - say 25% on time - average current draw should be something like 8 amps. I would expect the chassis alternator could keep up easily.
Would likely want some method of switching between the inverter and shore power - wouldn't want that 30 amp draw off your converter when the refrigerator is cooling on shore power - that would put a big dent in your converter's 55 amp capability.
Just some thoughts on the subject -