Registering if you have no rights is a criminal matter
Trying to register someone else's work as yours is a breach of copyright law as you do not have the author rights. In fact, it would be a criminal act, as I will show below.
But what can the damaged author do? Well, let's look at Smith v Summit... and sadly that is about wrongful DMCA takedowns, not illegally registering.
However, the case of Unicolors v H&M might be on point:
(a) The Copyright Act provides that a certificate of registration is
valid, even though it contains inaccurate information, as long as the
copyright holder lacked “knowledge that it was inaccurate.”
§411(b)(1)(A). Case law and the dictionary instruct that “knowledge”
has historically “meant and still means the fact or condition of being
aware of something.” Intel Corp. Investment Policy Comm. v. Sulyma,
589 U. S. ___, ___ (internal quotation marks omitted). Nothing in
§411(b)(1)(A) suggests that the safe harbor applies differently simply
because an applicant made a mistake of law as opposed to a mistake
of fact. If Unicolors was not aware of the legal requirement that rendered information in its application inaccurate, it could not have included the inaccurate information “with knowledge that it was inaccurate.” §411(b)(1)(A). Pp. 4–5.
In the questioner's case, the claimant (provably) has knowledge that the claim of owning the copyright is wrong, so if the certificate comes under review and everything is provable as the question claims, we need to assume it will be found invalid under 17 USC 411 (b)(2). If the Author thus proved the invalidity in a case, that would open a huge can of worms against the wrongful claimant. Why? Because it's a criminal offense to claim a copyright on a work you have no right to under 17 USC 506(e):
(e)False Representation.—
Any person who knowingly makes a false representation of a material fact in the application for copyright registration provided for by section 409, or in any written statement filed in connection with the application, shall be fined not more than $2,500.
That means the wrongful claimant not just has the copyright registration revoked in civil court, but could be prosecuted and adjudicated a convicted criminal for every one of those wrongful registrations separately.
Further, any money they got from prior settlements for which they used the invalid registration would become proceeds of fraud, since their false representation to obtain the registration which then is used to extract the settlements would constitute intent, and thus fraud, when the registration is found to never have been valid in the first place—and thus, they could get prosecuted for fraud in addition to the false representation, and is civilly liable for the damages.