Yes, BSD licenses are compatible with AGPL meaning BSD licensed code can be combined with AGPL licensed code while complying with both licenses. However, it does not give you permission to relicense the BSD code (or derivative works) as AGPL. The author is free to license any new code they write as AGPL, however the license for the machine translated code is another question. If it is considered a derivative work (which I think it should be) then it must remain under the Postgres license.
If it is not a derivative work, then for copyright to apply at all then it must be an "original work" which has "at least a modicum" of creativity applied by malisper in the translation. If this is satisfied then malisper could choose any license for the translated code they want, compatible with Postgres or not. If it isn't satisfied then no license applies, because it isn't eligible for copyright - essentially it is public domain.
The safe and polite thing to do is to keep the same license when performing machine translation.
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.