While travelling, the 120V is not in play unless you run your generator while driving, so let's focus on the propane operation.
The fridge does not care if it's moving or not, so if it works on gas while parked, it should work while driving.
As mentioned above, the gas valve does need 12V to stay open. You said you cannot connect the 12V from the truck (why?) but as long as your battery is charged, that should be fine - do other 12V items work? Lights, etc?
There may be a fuse for the fridge 12v - either in the fuse panel or on the fridge board - but again, this should be ok if it works while parked & not plugged in.
The only other difference I can think of while driving is the wind - maybe the pilot/flame is getting blown out? Are you missing any covers/shields around the burner area or fridge access panel?
Maybe taping a piece of cardboard inside the access panel to block most (not all!) of the vents will reduce turbulence inside the access area?
That's all I can think of at the moment. Good luck!
CC