Maybe it will help if I explain what the various components do.
The converter/charger converts 120vac input to 12v power, typically about 13.6v [Some converter/chargers have variable voltage algorithms for battery charging, but that's not relevant here.] The converter function supplies all 12v needs as long as 120vac shore power is available, and the charger handles battery charging at the same time. You say that everything works when on shore power and that the battery is receiving 13.5v, so we know that the converter/charger is working and that is is delivery charge voltage to the battery.
Your problem apparently begins when you disconnect from the 120vac. The converter can't supply 12v so the battery source is supposed to take its place. The wiring that does that is internal to the WFCO power center. Since you believe that the wiring to the battery is ok (as shown when 120vac is present), there may be a bad connection inside the WFCO. You can open it up and try to fix, or you can replace the entire unit.
This brief manual may help:https://manuals.heartlandowners.org/manuals/Electrical/Converter/WFCO/WFCO%20Ultra%20III%20WF-8900%20Power%20Converter%2055Amp.pdf