Here is what I think. Your converter is putting out full output. It's not getting to the battery. Put your multimeter on the converter output terminals and see what it says. It should say 13.7 volts. Put your multimeter on the battery. It should say the same as the converter output terminals.
If the two do not have the same reading, either the negative or the positive to the battery is loose. Run a jumper cable between the battery positive to the converter positive. Did it fix it? No? Run the jumper cable from the battery negative to the converter negative.
Hot means working hard. Not broken.
