The rules are simple, if perhaps irritating for you. You get four flight coupons in the southwest Pacific (ie Australia and NZ). You are using all four as PER-ASP-CNS-SYD-WLG. If you then want to fly AKL-SYD-HNL, the AKL-SYD sector counts as a fifth flight coupon in the SWP, so you can't do this. If you can then fly AKL-LAX via SYD on a single flight number (I don't believe you are correct on this: AA must have quoted you based on the nonstop AKL-LAX QF-operated flight, which is codeshared with AA and BA (I think), but assuming you are correct for a minute), then the fact that your flight would actually stop in SYD would be irrelevant since you would have just one flight coupon showing the flight as AKL-LAX, and this would not count against your limit of four coupons to use in the SWP.

There are two possible solutions if you want to go to HNL. The first is to cut out one of the other flight coupons from your ticket and buy it separately (so if you can find a cheap deal for CNS-SYD one-way on QF, for example, which is possible since most QF fares domestically do not require roundtrip travel, you could buy that separately and incorporate ...-PER-ASP-CNS//SYD-WLG//AKL-SYD-HNL-... into your LONEx ticket.) The second is to buy an AKL-SYD or AKL-HNL ticket separately, but I guess this will be more expensive.