Here too, I contacted Dometic to get the low down and the answer is that they design the units to be operated while in transit in any available mode, but I was specific to ask if it could be run in LP mode while on the road. The answer was a resounding "YES".
The newer fridges also have electronic ignition and are computer controlled, unlike older units that ran all the time. The newer units (in terms of Dometic) will light and shut down as the temp inside the fridge dictates need. They do not run all the time. If in fact there is a problem while on the road, the unit will shutdown and go into a trouble state, where a light will come light up on the display letting you know the unit is not operating, so it will not simply be pumping unburned LP into the area.
Computers manufactured by companies such as IBM, Compaq and millions of others are by far the most popular with about 70 million machines in use worldwide. Macintosh fans note that cockroaches are far more numerous than humans and that numbers alone do not denote a higher life form. -NY Times 11/91