Another member reported a similar problem a couple months ago. The problem with his trailer was that the refrigerator and the furnace were both tied to the same "12 volt supply line". When the refrigerator was operating, if the furnace fan came on, there would be a "voltage drop (negative spike)" that triggered the fault light on the refrigerator.
In his trailer, the battery was good, converter checked out, no problem with the refrigerator or the furnace, but when both were operated at the same time, the refrigerator would "throw a code"....
You might try watching the "timing of when it triggers". If the code and the furnace look like they happen simultaneously, try locating/rewiring the refrigerator to its own battery supply wire. That ought to fix your problem.
2015 F250 6.7l 4x4
2014 Cougar X Lite 27RKS