Welcome to iRV2.
The control panel on my Dometic fridge is 12v powered , and uses relays, and a circuit board, to turn on the 120v heater and 12v solenoids to control the propane.
Am I understanding your post correctly.
When you plug in to shore power the fridge won't operate because the control board won't light up ?
Have you checked the operation of your converter/charger?
What year is the fridge and RV?