One more thing you can do as a test is to leave the key OFF so that the turn signals will not work. Flip the turn-signal lever to one side or the other and then press the brake pedal.
If your one brake light works (whichever side you flipped the lever to) then I'd say it's a turn signal switch failure.
If it still does not work then you still have to follow the wires between the brake switch and the column connector. Then up to the turn signal switch if necessary.
Since this is an LED board, does it use the same elements for brakes as turn signals? Or do they try to break them out independently? if the latter, then there could be a fault with the board.
If it's just a simple plug-in and uses the same elements for both brakes and turn signals as would be normal, then the fault still lies up between the turn signal switch and the brake switch.
Full disclosure, I've never tried the bit about flipping the turn signals on with the power OFF to see if that would indicate anything. But it seems to me that it would work.
After all, if your turn signal circuit sends power to the rear lights, and your brake switch is getting power all the way to the turn signal switch, then your lamp should light up when you hit the brake with the signal on one side.
Another test to see if the circuit is good up to the turn signal switch is to turn on the hazard flashers. When you hit the brakes all the flashing should stop and the lights should remain solidly illuminated.
This would also indicate to me that the brake switch is getting power all the way to the turn signal switch.
Again, never used that to test a circuit either, but that's it's normal function when everything is working as it should.
paul