On the LHR-LAX-HNL segment, you will be required to collect your bags to go through customs and immigration. You don't HAVE to recheck your bags until the following morning when you depart. Whether or not AA will allow you to check them in is something you need to call AA about, but most likely they will tell you that you must take your luggage with you.

On the return, your luggage will be checked straight through from HNL-LHR unless you ask the agent in HNL to short check it. They will likely do it, but it may or may not be worth it to you... no reason to think that sitting in LAX during a regular day is somehow more risky than the standard TSA/employee pilferage.