I think what will happen is the fridge would go into lockout.
If it is set to auto (AC/Gas) and there is no AC present it will switch to propane mode. It will try and light (with propane) 3 times then lockout. At that point it will not operate in AC mode or Propane mode until it is manually reset (turned off then back on).
2007 Surveyor SV230 - 200 Watts Solar/MPPT Controller - 230 AH Battery Bank - 600 watt PSW Inverter - (2) 2000 watt Inverter Generators - LED Lighting
2009 F150 - 5.4 Litre - Tow Package
Boon Docking 99% of the time.