Just an FYI for any e53 owner curious as to the location of the positive battery cable run throughout the vehicle.
If you can gain access to the engine bay by removing the under panel...and if you can reach the starter or the alternator...you may be able to get a cable attached to the positive side, then find a ground...and you may be able to get enough power to the vehicle to unlock the doors electronically if you're in the same situation as the OP (malfunctioning driver's door handle thus lock cylinder that can't be manually actuated).