You have copyright immediately as soon as you put the piece on paper or the equivalent.
Registering copyright is a different matter. Registration gives you a solid timeline for purposes of originality and the right to sue for punitive damages and court costs rather than mere actual damages. There are provisions for registering anonymously, although that may mean that you are anonymous in the public database but still have to give your name on the original form. You can check at loc.gov. However, the odds that you will ever need to defend yourself through a registration are so close to zero that it’s not normally an issue.
Few publishers would ever trust an author whose name is deliberately hidden from them. It’s possible that you may find one willing to go this route but it places an enormous barrier on your work. You can have them issue checks to a corporate name, but my understanding is that setting up such a corporation is so expensive a proposition that only writers who have to deal with millions of incoming dollars ever do so. And your corporate information is a matter of public record so you could be searched from that.
The only practical suggestion I have is that you make such a huge name for yourself under some pseudonym online that you can approach people from a high point. I still can’t imagine why they would believe that you’re you without some proof, though. Perhaps you could find a trustworthy agent who could be a go-between. Publishers send royalty checks directly to agents who then take their share and send the rest on. A signature is necessary to make a contract legal, though, and I don’t know how to get around that. We don’t live in a society that makes anonymous business transactions easy, and for good reason.
This advice is good for the U.S. only. You don’t list any location information so you may not be in the U.S. The idea that our government would care about someone writing “controversial” opinions is so laughable to the 99.99% of the rest of us that it casts doubt over whether you’re an American.