I've been using a fake name and address at Namecheap for like a decade. Pay in Bitcoin.

I have a couple domains under alias names and PO Box

As mentioned Mynymbox or Flokinet both allow non-kyc accounts and crypto payments. Every hosting service or domain name service I've ever seen even with kyc has privacy protection (for a fee) so your real name isn't listed. Obviously the service knows, but not shared with the wider internet.

I do my domains through hostinger.com which lets you keep the registrar info private for no extra fee. They have a Bitcoin payment option too.

https://njal.la/