You need to watch out for the OWE rule that prohibits transiting your point of origin, in your case LAX.

That means, for example, that if you go LAX-LHR....MEL/SYD-LAX (in order to get on the A380, which has the only horizontal beds in business class for Qantas - the other aircraft have angled-flat) you wouldn't be able to go on to NYC; your trip would be over in LA. There's even a good chance that while QF 107 is shown as SYD-JFK, it often turns up in the PNR as two sectors, SYD-LAX and LAX-JFK. That might be a problem, plus QF 107/108 is with 744 equipment, so no horizontal beds.

The solution might be to begin your trip in San Francisco or Phoenix, both of which have BA nonstops to London, which would allow you to transit LAX on the way back from Oz.

As for the flat beds, BA, Cathay and Qantas (A380) have them in business. Opinions vary on the comfort level of the Cathay seat - mine is powerfully negative (they're coffins). Also they're on most Lan longhaul planes, but that's of no interest to you for this route. I think you'd be best by going SF/PHX on BA to London, short shuttle run to MAN and back, then BA to Bangkok, then BA to Sydney and Qantas to Melbourne, then shoot for the A380 from MEL/SYD to LAX and AA to NYC to end.