13 at the converter and 12.7 at the battery indicate it is charging a very low battery. As the battery comes up. both those voltages will rise until it is 13.6 at the converter and almost 13.6 at the battery.
Check the battery voltage with wires off. Connect the wires and turn off the converter and run the furnace fan and whatever works, now check the battery voltage--did it drop a bunch under load?
Meanwhile, check the DC fuse panel for the "lights" fuse. Something could have shorted a lamp on that circuit--water in outside light or a shower light can do that. EDIT forgot the lights do work with that switch closed. Getting mixed up!