I did camp at 10k' attitude and drove 12k+ with RV and my refrigerators work just fine. It was "some time" ago so the refrigerators were pilot-type that did not have electronic boards.
But still I don't buy the explanation you've got.
Propane burner is opposite principle to carburetor.
When in carburetor air sucks the fuel and mix ratio is crucial, in burner gas is sucking the air and mix ratio is having pretty loose tolerance.
Did you try to clean everything and see at what point the flame goes out?
Some advanced boards will flash a code.
Since the problem might be just relighting, how about setting the thermostat to lowest and let the burner work 100% duty.