This really boils down to stick-to-it-iveness and a bit of luck. As stated above, premium class seats are extremely rare. I felt like I'd won the lottery when I managed to score First Class round trip from Dallas to Sydney this past September. Those seats became available back in January of 2017 and I was fortunate to be able to get almost three weeks in Australia.
I might add that once you're there, getting around is much more easily booked via Qantas' 20000 mile Business Class awards. (Your award ticket to Australia will allow you to travel to one onward destination in Australia beyond your point of entry) I had no problem booking a few days in Alice Springs, Tasmania, Brisbane and Perth. Qantas' transcon Business Class product - especially on the A330s - is top notch, as are its lounges - probably the finest domestic Business Class lounges in the world.
Good luck finding those seats. Flexibility may very well be your best friend in this regard. Remember, if at first you don't succeed, try, try again!
P.S. I can't speak to Fiji Airways availability as I've never flown with them.