If you can adjust the routing/order slightly you can get it under 22,000 miles (excluding TPE). Or you can include TPE if you take the BNE-LAX flight instead of SYD-LAX (I understand exactly why you want to go SYD-LAX).

For example, lax-hkg-bkk-sin-syd-bne-syd-lax is about 21,530 miles. And lax-hkg-tpe-hkg-bkk-sin-syd-bne-lax is 21,740 miles.

The price goes up quite a but if you need to purchase the 26,000 miles option. I would price out the HKG-TPE-HKG fare as a separate ticket. Then you can decide if that cost is worth sticking with a SYD-LAX routing.