You don't have to "register" your battery with the ECU on a Z3. Forum folklore that's not true.
There are a number of signals the central locking needs to work correctly:
- Door position status, tells if the door is open or closed, both open and closed are required. This switch is on the body work side of the lock behind the latch pin. Check that the switch is moving freely. With the door closed, push on it a bit more and try the central locking. If it works then you need a new switch. Also try turning the plastic around on the latch pin as it may be worn and stopping the travel needed to activate the switch. If you have to change the switch, only fully remove one screw at a time. The backing plate is not fixed and will drop if all screws are removed. If this happens you'll learn how to take more of your interior apart than you would want to, just to retrieve the backing plate.
- Door lock status, both locked and unlocked are required. This switch is in the door lock actuator inside the door.
- Trunk locked/unlocked. Usual fault here is in the cable that goes from the body to the trunk lid. The wires break inside the sheathing.
From your symptoms, I would look at the passengers door, check the door position switch first.
Muzz