The BCM doesn't control the trailer light
connection, AFAIK. And as
@iamdub noted, there is a wiring block under the right rear quarter panel that connects the various lighting circuits for the back end. Might be good to check there once you confirm the fusebox is providing power.
The underhood fusebox has two separate fuses (L&R) for the TRAILER turn signals / brake lights (and separate from the truck body lights). To be clear, have you verified that those two fuses have power when the turn signals are activated?
I see that you've indicated that the trailer brake light circuit works -- those are the same circuits for the turn signals for the trailer. I would recommend testing the circuits with a regular bulb (more than just a test light) to make sure the circuits can handle the load. Something funny is going on here if the brake lights work but the turn signals don't.