Your wiring gauge is not the problem, especially not the short wire connecting the two batteries together. My trailer has 6 gauge wire between the batteries and from the charge to the battery about 20 feet away and it charges fine.
You mentioned you have a multimeter. Unplug the trailer and run the furnace blower for 3 minutes, then turn it off. Take a reading at your batteries, if you have the ability also take a reading at the circuit board for the converter charger. Then plug the trailer back in and take readings at both the charge controller and the batteries, and let us know what you find.
I'm betting you have a loose ground or a blown reverse polarity fuse. but you won't know anything until you take some readings.
