To me the whole notion that BA could step in because that would be useful to QR makes no sense. Ultimately, for a new route to work as a codeshare, it has to be of interest to both the prime carrier and the code sharing one, and I think that opening DOH-MEL/BNE/PER would make exactly zero sense for BA.

QR convinced BA to open flights to DOH from London as it adds capacity for them and BA are happy to have that destination.
QR convinced AY to open flights to DOH from CPH and ARN because AY had spare frames and crew due to the slow recovery in Asia and it makes AY quite happy to be able to steal some customer from SK.

By contrast, BA has no interest in flying DOH-MEL/BNE/PER and already too few frames and crews for its own desired expansion plans. Those are simply not the right routes for BA and BA are simply not the right partner for QR for that.