I live in BNE, I’m s Brisbanite so I can give you some advice.
As some have already said, if you are leaving after next June, then Air Canada is the best bet. LM charges 80K one way BNE to YVR and availability is quite good if you don’t mind departing on Tues. I’ve seen up to 3 seats in business class per flight.
You can forget about using points on Air NZ. It doesn’t exist. So that option is out.
QF either direct or via LAX will also be hard to find, although if you are willing to book 12 months in advance you shouldn’t have problem finding seats departing BNE. But you can’t depart on weekends. You can use QFF or Asia Miles. Alaska and AAdvantage calendar opens at 330 days, it’s full 30 days behind QFF and Asia Miles. So chances are many of these seats are taken.
Routing through Asia is easier to find seats as many have suggested. Cathay should be easy to find seats using QFF or Asia Miles. But it can be very expensive. AA won’t let you route through Asia so Alaska is your best bet at 60000 one way.
If you are willing to depart from SYD you have more choices. Just buy a cheap ticket between BNE and SYD.
Alternatively, you can consider buying revenue ticket. EVA from BNE-YVR is about $5000 rt. Not cheap, but consider the exchange rate is not so good, buying miles is not as good value as it used to be. So if you are less flexible in time or if can’t find award seats on the day you want, I personally would highly recommend EVA.