After being a few times to NZ (both Islands), I do think that your schedule is nothing but a big hassle. You end up seeing nothing and have the constant preussure to move on.

Is there any way of taking out either the South Island or the North Island? If not, then what are your main interests and focus on these. I'd also recommend to bypass Franz Josef and go to Mt. Cook or Lake Tekapo instead. And don't forget that the conditions in the Alps might be tricky with lots of bad weather.

One final advice from my side: Don't underestimate the jet lag and the fact that you end up driving at the "wrong" side of the streets. The combination of both can cause real problems.