Get as early as you can to BNE dom terminal at the airport, and explain, if you are lucky, they might put you onto any open seats, for earlier dom flight from BNE - SYD.
You will or should be able to do OLCI with the carrier, ie, it will be with CX from SYD.
www.%cathaypacific.com with your CX PNR, take away the percentage mark.
Main point with doing OLCI, is to let the airline know you are ready, but this is only part of the job done, still need to do bag drop and get passports swiped.
Then just need to do bag/s drop in SYD T1.
If QF will link bags, then lucky.
If not, 3 hrs can be quite tight, esp as you have to exit into public area at SYD T3, grab bags, grab a taxi, go across to T1, then bag drop, get passport swiped, and then go through immi and int security.
If it was all a linked PNR, then you would just need to show passport to be swiped in BNE.
But as the operating carrier is CX, passport swiping will need to be with CX.
As you have bags, you will need to have bag tags, so the CX kiosks would not do.
But if you succeed with OLCI, and can print BP, then its half the job done.
Make sure to still get BP from the airport staff, though.
Best of luck.