While I don't condone billing your interview at all, OP's experience still seems over-the-top for an interview, does it not? I can understand being frustrated with aspects of the feedback that could have easily been resolved in the first code reviews of full employment.
Don't get me wrong, I agree, OP went well above & beyond.
Even so, billing for that would get a response from legal, rather than HR (or me).
The lesson here, is that OP should have stopped sooner. Particularly on the first reply from them that suggested improvements.
I've had 1 person that I've given feedback to (because the project didn't compile!) - they were subsequently hired (against my advice). I had to fire them a week later.
If there is need for ANY feedback loop in the interview process that isn't face-to-face, it's time for both parties to walk away.
We're a place where coders share, stay up-to-date and grow their careers.
We strive for transparency and don't collect excess data.