Your description of the heat and associated fan not cooling the converter WFCO 55 amp is an indication the converter is at fault. Just replaced one in another unit which has been acting up for 2 months...worked some times, died with a load, in this case knocked out shore power and unit went on batteries.
3 or 4 lights should kick the fan on and stay on.
I doubt shore power would be good. Load it up and listen to your low voltage beeps and whistles. The battery may soon be discharged.
Just my thoughts after replacing a wfco 55 amp in a 5th wheel with option generator.