I'm travelling with family (4 pax total) from Christchurch to Sydney tomorrow with QF as part of a BA ticket - ultimately returning to LHR. There had only ever been 1 or 2 J avios seats available on this leg (now zero) however there are currently 7 seats for sale out of a 12 seat cabin. I'd like to upgrade (avios or cash) and spoke to BA who said it's entirely up to Qantas, who naturally won't do anything as it's a BA booking.

My question - is there any way that BA can / would ask Qantas to release more J reward seats? Additionally / alternatively, might an AUP be possible (again QF customer services suggested not)?

Many thanks.