Certainly got some good info there.
Starting with the running lights. That relay should have two small pins (85 and 86) and two large ones (30 and 87). Test for ground at pin 85, if no ground present, thats the first problem to fix. Second pin 86 should be at battery voltage when the headlight switch is turned on irrelevant of the key position, if not we are gonna have to look for problems back towards the headlight switch (not an uncommon problem) Pin 30 should be at battery voltage at all times, if not, you'll need to check fuse 8 40A, and the continuity between the two. Lastly, test the running lights pin on the vehicle trailer tow connector both with the relay out and also with pin 87 jumpered to battery positive. Another way to test the connection from that fuse until it breaks off for the lights, test for battery voltage at the red/orange wire on the trailer tow connector, if no voltage, check the wiring after the fuse for issues.
If all that checks out, then you'll either need to test or replace the relay.
The brake controller may have spiced into that 40A fuse and broke the connection back from there. I'd start by checking that out to see where they hooked it up, or if they used the factory brake controller harness.
My question, you said you get power to the running/clearance lights when you hit the brakes? What color wire is that? The brakelights wire should be light blue.