IANAL, but calling this "relicensing" is technically inaccurate. It is more precise to describe it as adding constraints. When you combine your work with upstream code, you are layering additional requirements (like copyleft) onto the existing attribution requirements. The original limitations remain in effect. Therefore, it is not a shift from A to B, but rather from A to A ∪ B.
This practice is entirely compatible with the PostgreSQL License, but it is often prohibited by GPL variants. You typically cannot combine GPL code with code under most other copyleft licenses, such as the Eclipse Public License.
Regarding copyright status, AI-assisted work is increasingly recognized as copyrightable in many jurisdictions, provided the process involves a sufficient level of human creative input (though the specific threshold varies by jurisdiction). Only work generated purely by AI, with no human involvement, is arguably public domain. In a case like this, which is akin to "pair programming," the output is almost certainly copyrightable.
IAAL (not legal advice) and I’m not sure the issue is settled.
The BSD license only explicitly permits the author “to use, copy, modify, and distribute this software and its documentation for any purpose, without fee, and without a written agreement.”
By default, the owner of a protected work retains all rights not conveyed to someone else. Changing the license isn’t one of the enumerated activities, and so I think there’s a case to be made that it’s not permitted.
Now if the author wants to claim it’s a new work, as opposed to a modification (which opens up a big bag of issues by itself because this was AI-authored), then the author can license it however they see fit.