Play with Google flights. Don't know your exact dates but using random ones at the end of June the following priced at $6k but requires an indirect to SYD (not sure what your price range is but that's obviously way less than 10k):

SFO - AKL - SYD (NZ J)
SYD - SIN (BA J)
SIN - NRT - SFO (NH J on 77W so lie-flat for both legs)

Or

SIN - SFO (new UA J direct on 787)

Whole thing shows as bookable via UA directly from Google flights even with BA for the SYD - SIN flight.

Seeing other options in this price range but all require indirect on the SFO - SYD route