The practice is that the baggage fee rules of the airline that checks you in (i.e. operating carrier of the first segment) apply; they will probably waive their fees if you are prepared to show that according to the rules of the carrier that sold you the ticket you are exempt from paying any fees, but YMMV.

There is no written "rule" about this recent practice, though, and AA.com is completely silent.

In your case, I'd call AA to ask; if they say that you're exempt from the fee at LIH, then ask them to annotate the record in a way that HA sees that annotation. Given that AA doesn't sell LIH-HNL, I have a strong suspicion that the LIH-HNL is a voluntary downgrade on a through first-class fare, which would be a strong platform to argue that fees are not due as you're flying on a first-class ticket on that segment; if that's the case, then you might also ask AA whether any LIH-HNL first class seats have opened up.