There are many alternatives as pointed out by mk4 so shift your business with them and avoid KYC exchanges at your will. But a lot depends on your local jurisdiction. If crypto is legal in your country, then you are good but tax will come along the way. But if the legal status of crypto is still not finalized, it's best to avoid kyc exchange.
Also look for a data protection law if available in your country. If you reside in European Union, the GDPR law empowers you to take control of your data. So you can ask the exchanges to delete your data.