Can you de-pin the wires in the regulator harness and swap wires or cut the wires upstream of the connector, cross them and re-solder? Is this a BMW part of an Ali-Express version? I'd avoid making wiring changes on the vehicle side of any connectors to avoid confusion if the regulator needs replacement in the future. Basically follow your indications to make this one work since it seems to only impact the regulator on that side.