Hi did you find a solution to this as my BMW 320ci E46 is doing the exact same

Hello, First, make sure the lights on in the rear are indeed the brake lights, not just the tail lights. There is a good chance that the headlight switch is causing this condition. Have someone press the brakes so you can see if the lights become brighter. If they do, manipulate the headlight switch and dimmer switch for the interior lights to see if there is any change. Also, the ignition switch may be staying on, even though it is turned off, and you can test the electrical portion without removal by accessing the wires on the steering column or testing the signal at the body control module. As you can tell, this may be a more in depth repair, so if the headlight switch and dimmer switch are in good working order, I recommend running this by a professional so you don't end up throwing parts at it.