Battery not Charging

It sounds to me like there may be a bad or broken connection in the trailer wiring between the truck plug and the trailer battery, as you've apparently verified things on the truck side pretty well. It should be a theoretically simple job to track it and see where the charge voltage disappears (though it may be practically rather more difficult due to tricky wire routing). There likely is a fuse or circuit breaker in the trailer somewhere near the battery for this connection, and possibly it is blown--or possibly the charge wire is not actually connected to the battery.

It would also be good to verify that the trailer ground connections from the hitch connector, and the corresponding ground connection on the truck, are good and solid, though a bad (enough) ground would also affect the trailer lights and brakes.