I think the heater element is both shore power and battery. So to run when plugged in off of 110 need to have the heating element.
I have a 77 Scamp with a traveler brand refrigerator, there was an after market company that had replacements. Seem to recall around $75 maybe? I had to use some Google foo to find them but then I did not have a part number.
All of the Dometics that I have worked with, that were three way, had two separate heating elements, one for DC, another for AC. Otherwise they would have to add a transformer to run a single element on both 12 VDC and 120 VAC. Other brands may differ.