Hi, you'll get better advice if you let us know what part of the island you'll be living in. If you'll be in the west, I highly recommend Dr Sophie Lacaze in St Paul - she's amazing and was very supportive of me aiming for a low intervention, natural approach.
Remember that Reunion is part of France therefore there are top notch clinics and hospitals all around the island to European standards. As for the 'best private clinic' on the island, that's a personal thing but I've had several friends tell me to steer clear of clinique Jeanne d'Arc.