I'm not sure if the BMU is smart enough to cut power to the brake light when failure is detected, but it might have.
Just to clarify one thing, did your running lights also stop working the moment you tested the brakes? Did you test only with the front brake (then it failed), or with the front then the rear brakes (then it failed)?
btw, Did you disconnect the battery, undo all grounds and disconnect all electronics before welding? If not, I hope they survived the ordeal.