B772
I was once on a Malaysian 772 which had a plaque on the bulkhead stating it held the record distance for an ETOPs aircraft. It flew its delivery from Seattle to KUL direct, non-stop, which calculates at 7016nm direct, but I would guess would be at least 10% longer. Of course this was neither scheduled or revenue generating.
BTW MEL - LAX - direct with which airline? United stop at AKL; SYD to LAX is direct with QF, NZ and UA, and there can't be much difference in that distance.