Will you use that fridge on propane and SLEEP in rig at same time?
There are more safety features & designed control functions in that propane system in your trailer then in the fuel system in your vehicle.
Propane system in your trailer is only 0.4 psi, has shut-off devices in the appliances (flame proofing/lock out failure), excess flow devices etc. PLUS the LP Detector Alarm.
Operating while in transit is part of what 'self-contained' is about.
And it is safe to do so.
If not I wouldn't use it while stationary and sleeping!!!!!