It sounds like the brake controller is working and you have a problem with the rest of the trailer wiring harness. I bought a truck because my 2008 E-350 could not be set up to tow even a small trailer without spending a fortune. The story I got was that a rear body computer is different for towing due to the electronic stability control. US models were not equipped with the computer that allows towing. It could be installed for about $1000.
My best guess is that the car's computer is detecting excessive current on the taillight and turn signal lines and turning them off.
The clearance lights operate from the same line as the tail lights. Tail lights will not be on during the daytime on automatic.
Back in the day there was a thing called a trailer isolator. It connected to a
12V line and the tail, turn and brake lights. The tail, turn, and brake light signals operated a relay or solid state switch that used the
12V line to operate the trailer lights. This limited the current drawn from the vehicle light wires.
Turns out they still make them. Maybe your dealer can figure out how to wire it in.
This one for a motorcycle trailer might work, or there are probably others with higher current capabilities.
https://www.amazon.com/Rivco-Univers.../dp/B0058P2QV8
Edit - looked a little further. This may be what you need:
https://www.etrailer.com/Custom-Fit-...ins/46255.html
Al