Yeah, Namecheap's been cracking down hard on fake registration data lately. If you manage to get back in and transfer out, you'll want a registrar that doesn't hassle you with ID checks.
Njalla is a solid option—they register the domain under their own info, accept crypto, and don't ask for your documents. Pricey, but private. OrangeWebsite (Iceland-based) is another one known for being lax on verification. Some people also use Porkbun with real info + privacy protection and call it a day. There's also Njord from Sweden if you want another privacy-focused choice, but same deal—not exactly cheap.
No registrar is 100% anonymous though. ICANN rules mean they can technically demand verification anytime, especially if the domain gets reported or involved in anything shady. If the domain is important, honestly might be safer to just register a new one with real data somewhere reliable and move on. Less stress, and you don't have to look over your shoulder every time they change their ToS.
Just keep in mind that hopping between "fake data friendly" registrars is a cat and mouse game. Eventually they either get bought out or start playing by the rules. Njalla's model is probably the most future-proof since they own the domain legally, but you're putting trust in them instead. Pick your poison.