As umguy said, it may be easier for you to get a reward ticket with CO Mike through Guam and into Cairns. The connection times for those flights are actually very good, but you would have to fly RDU-IAH-HNL-GUM-CNS-BNE. That's a lot of connecting, RDU-IAH-LAX-BNE looks a lot better. I haven't had any experience with QF rewards from CO, but everything that I've heard supports the above claim that Qantas only releases two business seats per flight for CO mileage rewards.

You also have the option of using a mix of DL and KE. You could see if RDU-ATL-ICN-BNE is doable with CO miles, but it might get pretty pricey. A third option would be an award on EVA through TPE.

Unfortunately, those three airlines (QF, KE, BR) are the only three Continental partners that fly to Australia westbound. If you really don't mind flying, you could also try EK's service JFK-DXB-SIN-BNE or JFK-DXB-SYD and then buy a flight to Brisbane.

Best of luck to you and please keep us updated as to how it's working out, I am saving up my OnePass miles for a trip like this too.