A few people have asked whether registering a copyright is even worth it for this kind of work. General points, not legal advice, and this varies by your facts and jurisdiction.
The Copyright Office has been clear that purely AI-generated material isn't registrable, but a work that combines AI elements with meaningful human authorship can be, and the practice is to disclaim the AI-generated portions and claim the human contribution. So your registration protects your selection, arrangement, and original additions, not the raw generation. That's a narrower right than a lot of people assume, but for genuinely modified work it's not nothing.
If you're doing this at any real volume or for clients, it's worth a short conversation with an attorney about how to document authorship and what's actually worth registering, because the answer really does depend on how much of you is in the final piece.