I would fly QR/EK if I absolutely had to fly LHR-AKL in July – HKG/NRT/SIN transits potentially might be on the table by then, too.
Transiting the US would be a last resort option IMHO.
Also, IAH/ORD/SFO won't be operating for a long time, it'll just be LAX-AKL for the foreseeable future.