But is there anything stopping a human from applying for copyright in their own name? Does the fact that somebody can recreate the prompt invalidate their claim?
Filing isn't the gate, registration is.
Copyright Office requires you to disclose AI involvement and disclaim the AI-generated parts. Zarya of the Dawn is the example — applicant filed for the whole graphic novel, got partial registration on the human-written text, refused on the Midjourney images. The reproducibility of the prompt isn't really the test. The test is whether a human made the expressive choices.
What you're asking is, "could someone do fraud" and "would being found out invalidate their copyright". To both of which the answer is generally, yes.
It'd be a form of plagiarism, just with different consequences to the most common form.