I can get it to work a bit (without booking).
- MEL-(QF)AKL-(QF)LAX-(BA)xLHR-(QF)MEL for LLONE4
- charged via QF AUD3499+AUD868
- LAX-(BA)xLHR-(QF)MEL-(QF)AKL-(QF)LAX for LONE4
- charged via BA USD4400+USD565
- LAX-(AA)xLHR-(QF)MEL-(QF)AKL-(QF)LAX for LONE4
- charged via AA USD4400+USD332
With the "BA" booking, I was asked to contact BA to book, the "AA" and "QF" booking, CC details were requested.