Thanks for all the answers.
So, I will give it a try with AA.
I know also a number of brick-and-mortar TAs, but I am not sure how long they'll be around - they are not available 24/7.

One last question:
hel-lhr-lax-jfk-dfw-hnl-hnd-syd-mel-per-hkg-nrt-pvg-hkg-dps-doh-hel

So the HNL-SYD routing via HND is no problem? I am asking again, because I am touching Asia twice.