I am unaware of any BCM having internal circuit beakers that directly control the output to various electrical circuits. A BCM is like a traffic cop and directs communication signals to other devices to switch items on and off outside of the BCM. All electrical circuit current draw is regulated via the vehicles fuse panel (whether inside the cab or in the engine bay). I am no expert on the RAM 1500 BCM, but most all BCMs working in this way.
I would recommend checking the lights, fully disconnected from the vehicle, on a 12VDC power supply to see if they, themselves are functioning properly. If they have been proven to illuminate without any interruptions, I would connect them to either the factory bed light wiring (if so equipped), or run the wiring into the trailer connect (as you suggested) and buy the following to CURT harness to connect the power and then mount a weatherproof switch in the bed to activate or deactivate the lighting:
If you want to get just a bit "complicated", you certainly can use a weather resistant relay to route the power to the bed lighting from the CURT when the wire feeding the power from the cargo light is activated and energizes the relay. Several ways to skin this cat, but there is certainly something going on with the power source that the OP tapped into assuming the lights themselves are working correctly.