My best one (trip) to see the Southern Alps snow, was to fly the EK 380 SYD - CHC in J on the A side, thats past the wings, and as the plane flew over, (this was about 2015 or so), and it was the best view.
Have also done F on that route.
This time round, I thought sitting on row F on NZ from MEL - CHC, that I could have a look, or try to see Mt Cook, but no luck, the capt on that flight said due to cloud, nothing much was visible, which was true.
I tried to strain my eyes to see Mt Cook, but no luck.
So, OP, if you have another chance, book that EK 380, and pick to sit on side A, either J class or F, and as the plane comes in lower, of course it depends on ATC and the capt, but that one trip was one of my best so far.
OT, planning to do Japan one Dec, maybe go via QF SYD - HND, and then go to Otsuku, outside Tokyo for the real snow experience.
Yes, its not cheap in J, but JL premY OR JL via MEL (to NRT) is not that $$$.
Timings also suit me if I finish work by 4.30pm, get to MEL, then get the JL flight to NRT, dep after midnight , arr a bit after sunrise, sitting on the right side will have a good view of sunrise.
Will end my diatribe here.