Mixed results
I promised I'd report when trying this trip again and have just got back home so this is what happened:
Trip booked with AA and the return trip was LHR-LAX-HNL-ITO with an overnight stop at LAX. Flight from LHR to LAX was BA, LAX-HNL was AA, HNL-ITO was Hawaiian - all AA codeshare partners. My bag was checked in all the way to Hilo (ITO) by BA at Heathrow despite the overnight stop. (note - I checked in online with BA but did not check in for the HNL or ITO flights at that time).
At LAX I took my bag to the hotel but didn't remove the tag. The next morning, I checked in using AA's self-service machines and clicked on "no bags to check in" with advice from an AA agent (this worried me a little) and just handed it in to the TSA bag drop off which they accepted without a word.
My bag didn't show up at Hilo when we landed. Fortunately, it did show up on the next flight from HNL and was delivered to my home by Hawaiian.
So, I'm not sure what to recommend. This is the first time in roughly 30 trips between Hawaii and LHR that my bag didn't turn up so not a good sign, but it did turn up shortly afterwards and I suspect the delay was at HNL rather than LAX simply due to the timing of the bag's arrival, but could be wrong. I definitely don't recommend leaving the bag overnight with AA at LAX plus I needed it anyway for the hotel stay - too many things could go wrong. Simply leaving the tags on and handing it to the TSA worked but not without some worry when the bag didn't turn up.
It seems the safest way is to tell the BA agent to check the bag only as far as LAX and then check it again yourself in LAX the next day but pay the $25 fee (or negotiate it away as I did last time which is a bit of a pain).
On an unrelated matter, I witnessed some exceptional service by an AA flight attendant during the LAX-HNL leg which motivated me to write to American praising her. You don't see this too often on AA but she was exceptional and hope she hears how much she was appreciated!