In theory after customs you could re-check in your bags to HNL and then go to the hotel, since your connection is under 12 hours, but LAX is one airport I really would not recommend for this. But if you did want to do this you would need the agent in presumably BRU to label the bags to HNL. You would still need to walk the bag past customs but there is a special belt after customs for reloading bags back into the system. The chief merit of this is that it's one thing less to worry about on departure morning and you would be HBO (hand baggage only), the chief downside is that it would increase the chance of the bag not getting to HNL on the same flight as you. Whereas if you collected for overnight I can't see that going wrong.

Next morning, well on Fridays and holidays LAX security can be the slow point in the mornings, otherwise it's reasonably efficient. Apparently it has improved, but if you don't have status or priority you best get to LAX 90 minutes before departure, maybe a little less for you since you should at least get the Priority channel being Sapphire. If HBO 75 minutes is OK, I'm comfortable for 60 minutes. The deadline for AA is 45 minutes but you don't want to risk it to that degree. On the other hand T4 (+T6, its sister terminal) at LAX is surprisingly small, though congested, incidentally. There is an Admirals lounge in T4 for you, upstairs half way down the corridor on the right, not massively signposted but if you look for the 3 stars logo you'll find it. This all assumes you either have boarding passes from BRU or you got them online or via the BA / AA apps. If you don't, add another 5 minutes or so.

The one thing to watch, depending on your hotel since they are all different, is the shuttle frequency and congestion. Getting to your hotel for the first time at LAX is a bit of an experience when the barmy (and polluted) air of Los Angeles hits you. On the return your hotel may operate to a timetable (or not) and if you turn up at the key morning period at the wrong time you may be asked to wait for the next shuttle, which may end up being stressful if the clock is clicking down.