I feel a little ignorant here, but:
- why would it matter either for QF or BA whether you flew LAX-JFK on AA or QF?
- what do you consider the benefits of reaching top tier on all 3 airlines? I can understand AA and either QF or BA (for AC access deomstically in US) but both?
To answer your question - I have not done THAT itin

but at one point I questioned QF whether I could fly SYD-LAX - stop - and then continue on QF to JFK. They said no problem. Based on that principle I can't see why you shouldn't be able to fly LAX-JFK in QF. But don't quote me