Do I need to have genuine BMW Motorrad M Carbon wheels entered in the vehicle registration papers, or for which models are the M carbon wheels approved?

No. The wheels are fully homologated.

M Carbon wheels are approved for the models S 1000 R (from model year 2023 onwards), the S 1000 RR (from model year 2019 onwards), the M 1000 RR, the S 1000 XR (from model year 2022 onwards), M 1000 XR and M 1000 R. As retrofittable optional accessories, the M Carbon wheels are available only for the S 1000 R (from model year 2023 onwards) and S 1000 RR (from model year 2019 onwards) models.