Not sure, but does the inside fridge have a switch to tell it what "fuel" to use? I have an old slide in camper with a Dometic fridge in it. It is a 3 way fridge (12 vlt, propane, 120vlt). I have to manually turn a knob to select which one it will run on...Just a thought.
Also, Something that I had read in one of the camping forums, If you have a Thermometer with a remote sensor...Example below.
typical thermometer with sensor...there are others to choose from.you can put the sensor in the fridge and have the readout portion in the truck (or house).
Just another suggestion...use at your discretion.