You say the pilot light comes on, after a short 'warm up' period, does the the burner come on? If not, you might have a plugged nozzle. Remove it and clean it out, being careful not to change the size of the orifice. Under 120 v, does the element get hot? Even though the fridge is LP and 120 v, it needs 12v DC from the battery for the electronic control panel to work, make sure 12 v system is charged and working.
You won't hear any fans when operating, it is pretty silent. It also takes overnight to cool down, they aren't fast cooling like a compressor unit.
Bob & Donna
'98 Gulf Stream Sun Voyager DP being pushed by a '00 Beetle TDI