If you have double the points of what is needed, I would do the following:
Secure the booking on QR or BA.
Keep looking for availabilitirs on Qantas. If something comes up, book Qantas, then once ticketed, cancel QR or BA for free. Cancelation of international awards is free and fullt refundable till the end of the year if not mistaken (check first).
If you dont have enough points, you can, instead of booking-canceling, call Qantas to change your QR or BA award to the Qantas one (if you find award of course). There is a tiny risk that the call center stuff up, as seen in several threads recently, but in theory that ahould work. This is allso free till the end of the year if not mistaken.
In any case, if you book QR, be aware that Qantas needs to ticket your itinery within a day or two (can t remember exactly) or QR will drop your award. So if you don t get the pdf ticket quickly, you will need to call and insist with the call center. BA is better as far as I know, in that aspect.
Good luck