Is there a way to reverse the damage? Perhaps through things like mixers etc? What are your thoughts on those?
Probably the only damage was that you have submitted your personal information to the exchange. For now, withdraw your funds outside of that platform, keep them in a wallet whose private keys are under your sole control, and, as mk4 suggested, contact the exchange and request that all personal information pertaining to your account be deleted immediately.
If you wish, you may also use a mixer. It's up to you, but it wouldn't reverse the damage. It wouldn't erase your personal information stored somewhere.