Originally Posted by
hrnt
Thank you all for your comments. I talked with guest services and they assured me that BA would be able to route the luggage all the way to LIH. I'll post an update if something strange happens instead

Well, nothing strange happened but I'll still post an update (so this post may serve as a data point for other people in a similar situation)
When I arrived at LAX, I contacted the BA representative at baggage claims. I received a 200USD prepaid Visa Electron as compensation (*) and I was promised that the baggage would be delivered to my hotel (in Kauai). My bag finally arrived about 2 days late, so all in all, ^ for BA.
I am currently sitting at LAX waiting to board the last flight to the LHR winter wonderland (and continuing to HEL on Sunday), so I guess the return trip might be even more exciting
(*): The Visa Electron was accompanied by a slip that referred to EC 261/2004. I didn't know that EC 261/2004 ruled on delayed baggage as well?