Why bother checking 12V... He said the burner lights up if the power to the fridge is on.
If burner ignites, then 12V supply is there. If fridge still doesn't cool, can be bad cooling unit or bad burner (ignites but the flame isn't good). Check the flame shape and color, plenty of videos and pictures about this. Bad propane regulator is also possible - stove would work but the fridge wouldn't. Get another repair quote, to knock a few hundred off that first estimate. I'm not sure that this tech even inspected it properly.