Unlike GPL, BSD and Apache licenses do not claim to also cover your non-AI-generated code that only invokes the AI-generated code.

However, even if the BSD/Apache/MIT licensed code can be incorporated freely in your application, you still have no right to remove the copyright notices from it and/or to claim that you own the copyright for it.

Therefore, unless the AI model has been trained only on non-copyrighted public-domain code, incorporating the generated code in your application means that you have removed the copyright notices from it, which is not allowed by the original licenses.

There is absolutely no doubt that using an AI coding assistant works around the copyright laws, but it is still equivalent with doing copy and paste with fragments from copyrighted works into your source code.

I consider that copyright should not be applicable to program sources, at least not in its current form, so reusing parts from other programs should be fair use, but only if human programmers would be allowed to do the same.