Suppose the user could claim ownership of the prompt used to create the work; unless that was also created using an AI.

Again this is the difference between authorship and ownership. While I'm no lawyer, to my understanding you must demonstrate that what you did was genuinely transformative to claim authorship.

You use compiling the code an LLM gives you as an example of this. Does a person running gentoo have the ability to claim authorship of all the software on their machine because they compiled it?

I think the only thing you can certainly claim authorship over is the prompt(s) that were used to generate the output.

Another good example of this is compilers. How often do you see people claim authorship of the assembly generated as an intermediary step of the gcc?