You cannot (*) use LLMs to generate code that you then license, whether that license is GPL, MIT or some proprietary mumbo-jumbo.
(*) unless you just lie about this part.
You can't copyright a work that is only generated by a machine: "In February 2022, the Copyright Office’s Review Board issued a final decision affirming the refusal to register a work claimed to be generated with no human involvement"
But human direction of machine processes can be copyright:
"A year later, the Office issued a registration for a comic book incorporating AI-generated material."
and
"In most cases, however, humans will be involved in the creation process, and the work will be copyrightable to the extent that their contributions qualify as authorship. It is axiomatic that ideas or facts themselves are not protectible by copyright law and the Supreme Court has made clear that originality is required, not just time and effort. In Feist Publications, Inc. v. Rural Telephone Service Co., the Court rejected the theory that “sweat of the brow” alone could be sufficient for copyright protection. “To be sure,” the Court further explained, “the requisite level of creativity is extremely low; even a slight amount will suffice."
See https://www.copyright.gov/ai/Copyright-and-Artificial-Intell...
The short answer is that it's possible if the prompt has sufficient control but only the parts controlled by the human are eligible for copyright.
Using AI doesn't automatically disqualify from copyright protection though.
But it will be a shitshow either way.
It's not clear to me how much code you would need to modify by hand to qualify for copyright this way, but that's not an impossible avenue.