As far as I can tell, this is a bit problematic.
You don’t have enough Emirates or Velocity points for much (though you could use your points to fly into Sydney where there are more options).
You could probably do something with your Qantas points, but not if you want to go business class via Singapore. There are four airlines that fly to Phuket from Singapore: Singapore Airlines (actually SilkAir), Scoot, Air Asia, and Jetstar Asia. You can’t use Qantas points for the first three. As for Jetstar Asia, I don’t think it has business class.
So your options are to fly economy into Phuket from Singapore (though of course you could use Qantas points to fly business into Singapore) or to fly via another city, or to pay.
My strategy would be to use your Qantas points to fly to Singapore in either business or economy (I might be wrong but I don’t think you have a PE option out of Brisbane), then use your travel credits to pay for cheap economy flights on SilkAir or one of the budget airlines to Phuket.
If you really absolutely have to use points to fly business or PE all the way into Phuket via Singapore, my strategy would be to spend a year building up your Velocity or KrisFlyer balance thanks to you and your wife signing up for, say, two credit cards each with hefty sign-up bonuses … and then go next year.
I’m not sure if this has helped but this is a tricky one!