Which Island in the South Pacific to Visit from SYD?
As others have hinted ... sounds like CNS is good match for your situation (except its not a Pacific Island) and CNS is probably more realistic based on your limited time and additional expense required.
If you're getting to SYD from NA with UA miles and haven't ticketed, then don't forget about the free stopover for UA awards to Aus/NZ ... you just pay the additional taxes, for example:
LAX-SYD-RAR-LAX
LAX-SYD-AKL-HNL/RAR/NAN/TBU/APW(TBU)-LAX
I've got WAS to CNS with a stopover in RAR on the way back ticketed with UA miles ... in C BTW
. The stopover is a great benefit many folks forget about.