jenn,

u know the rules well
the only way to do it is to shift your HKG/NRT to be together with the SIN trip


i.e. ZRH-LHR-(USA)-LAX-SYD-(Oz)-HKG/NRT/SIN-(+2 segments)-LHR-ZRH
i guess there is no other ways

you could save segment by, e.g. SYD-NRT-SIN-HKG-SIN-BKK-SIN-LHR.
CX has (i thk) one daily flight BKK-SIN