So, 1 hour 22 minutes on the phone to Api in Fiji.
The missing flights are now in my itinerary showing as Pending. This means they are not confirmed. Api says he has contacted the airlines seeking reinstatement of these flights. Api said specifically and more than once that he (i.e. Qantas) guarantees that the flights will be reinstated and ticketed. I pressed for a timeframe and was told after some difficulty that they would be fully ticketed within 72 hours.
I asked for a copy of the recording and was told no, despite multiple requests. I was told that a copy of the recording is stored against my booking reference number. Yeah, right.
There is not much I can do for the next 72 hours, it seems.