Assuming that the lights are powered by a normal 12 volt supply lead you could run a wire from a 12 volt source like the battery and connect it to the connector for that light, to one of the lights in the rear and verify the light works ok that way. That would prove out that end of the circuit at least. One thing to be cautious of, the brake lights may be tied to some sort of control module that is in between the brake switch and the lights and it is causing the trouble. I don’t know for sure, but with today’s designs of wiring systems I wouldn’t be surprised if that was the case.
Another possibility is a problem with the backup light circuit.
“A few days ago I backed the car up to the front of the house, and in the reflection of the windows and glass I noticed my brake lights were very dim”