Thanks for that.
I checked out the pricing for F-class one way and it's several hundred dollars more than my option. It would be nice, but I really can't justify it.
I have to assume, on the basis of no comments to the contrary, that my suggestion is the most cost effective way of getting 240 status credits, departing from an Australian airport, all costs being considered.
The $950ish option is a pretty good one if you can actually book it that way. Of course as you note you have to get to CHC.
What I'd be doing is looking out for the (almost monthly) EK sales ex NZ where they can give you a one way F fare for ~$NZ900 or so
one thing I'm a little confused about - what is the goal here? looking at your profile you are a 2MM on UA (I feel for you

) and an EK Gold which trumps QF Silver. From the context if your post you want to go to QF Gold? What about the EK status? do you want to keep that too or move? I think given the EK status it may be better to work towards requalifying there? but obviously it's your goals.
Anyway I've been seeing these ex-NZ sale fares it seems monthly, but for short periods. It is the best way in this region, without bouncing around Asia or the USA for "cheap"(relatively) SC's.
Your other option is a return SYD-AKL/CHC in J fare for 160SC+50%=240. However the last decent sale ended recently, and even that was more than your budget (as you say) at around $1300ish return.
My only other thought is that given with your plan above you'd need to get to CHC that's an extra couple of hundred $. So your total would be, let's say $1200-$1300, and you may find something like a CBR-SYD/MEL-OOL return in J may be just a tad more(sadly you just missed the 95th anniversary sale) but would still get you 160SC without the passport use and could do it in a day.
Your other options would involve flex Y fares to things like CBR-SYD-DRW or something which would not be as much fun.
good luck but as 11sjw notes, it may not be worth it to you to attain QF gold, given your EK status and potential future travel, which going purely on your profile would seem to be spread out a bit.