While I haven't tried anything along this line, this thread caught my interest. From what I can tell, the Dometic 120 volt heater elemenmt is just 325 watts: Dometic 120V 325W Refrigerator Heating Element : Amazon.com : Automotive
. At that wattage, even at just 90% efficiency, an inverter would be drawing around 30 amps. As the fridge would only run maybe 10% or 20% of the time, not a big alternator load while driving.
Also, since the element is a simple resistive load, I believe a cheap "simulated" sine wave inverter should work fine ("true sine wave" not necessary in this application).
Would be nice to wire it so the inverter is only connected when the vehicle engine is running and there is no 110 volts from either shore power or the onboard generator. Sort of an "auto-switchover". I can imagine a circuit to do this, but easiest is probably "manual" - unplug the fridge from its normal 110 supply and plug it into the inverter.