You have a 2010 Forest River toyhauler trailer so that 2 way fridge uses 12V DC power for the control board whether using 110V AC electric heat element or the LP propane gas mode.
On propane it also uses 12V DC to power spark electrode & operate gas valve.
There is NO pilot flame......that IS the main flame. The control board will open/close gas valve and spark/re-spark to light main flame based on fridge thermostat setting.
On shore power the converter will supply the needed 12V DC.
Without shore power the trailer battery will supply the needed 12V DC
When connected to tow vehicle, the trailer battery will supply the needed 12V DC and tow vehicle alternator will maintain trailer battery.
Fridge places very little demand on 12V DC System.
Furnace is a DC hog, electric stabilizers/tongue jack high demand but short duration, lights next, water pump when running, water heater/fridge come last (control board/gas valves)