Ok, thanks for all the input. Our flight is BRU-LHR-LAX-OGG, so LAX will be our first entry to the USA. Global Entry probably won't be an option, since it too much hassle for this one-off trip.
I do think I won't have expedited security fast-track with OW sapphire (not from BA but AB), isn't this a OW Emerald benefit? Perhaps BA offers this to their own elites?

The hotel is the Travelodge hotel (perhaps some of you know it). On the return I've booked the Hilton since we have more time. We'll make sure we leave at 5 AM in order to stay clear of any problems with long queues at LAX. Are the queues for domestic flights long at LAX, meaning is there any sort of security-checkpoint for visa's etc which can take a while, when departing again?

Thanks all!!