Setting aside the -356, if you're flexible on the dates then baredemptionfinder does a good job at finding the remaining flights if you type in SIN -> SYD. There's more Business/First seats to/from Sydney than you might think - it's not a total washout.

You might be able to combine this with a BA11/12 flight from LON -> SIN a couple of days earlier, or even a BA33/34 flight from LON -> KUL and a short connecting flight (although your companion voucher would then end in KUL).

If you need -356, i'd aim to bag LON -> SIN or LON -> KUL first. The second leg is shorter and there are lots of cheaper options if you're unable to get the elusive SIN -> SYD on the phone the next day.