I've actually done this. Its an art of social engineering that I call how to get your way on the internet without provoking a Streisand effect.

The company being registered doesn't matter, I don't expect people to search through 250+ jurisdictions around the world to figure out where its registered. So you can pretend the company name is the same as the domain name, reference "Your Client" in the sternly worded email, and "trade secrets" or "unauthorized privileged information" being hosted on "your server"

A lot of companies will take it down very quickly and apologize profusely. Anywhere in the world, US, Russia, you name it.