You can add another stop on this trip via the US east coast, to increase the no. of BA TPs to 1,120. E.g. CAI-LHR-JFK-LAX-HNL. Price in Aug is as low as £1,239.

The AA fare basis is INE0F6RS and requires a min Sat night stay. There's an equivalent BA fare basis which is IRCME but that's more restrictive - min 3 night stay.