the 12 VDC is not the 12VDC in the BMW world... BMW is using so-called PWM in its LCMs, Pulse Wave Modulation, where the full 12 VDC is applied to circuitry in waves of 12V-0V-12V-0V, thus effectively reducing the illumination of the halogen light... if you connect an LCD light to the same circuit, you will see rapid flashing of the LCD as the response time of the LCD is much-much faster then a halogen light... If one light is really dimmer than the other, it might have too much of the PWM signal (wider gaps of 0V, dimmer the light)...

Since I do not know why the light will do one side in PWM and not the other side (or, apply different values of PWMs) we need to make sure that this is a really case of the PWM applied to your lights...

do you have 12V leds floating around, that you could have connected to the light, for the test purposes?