Ask for New Zealand Driving advice

I will go to New Zealand in the middle of May. It is about the winter. Do you think it is too dangerous to drive on South Island? Will the road be icy and slippery?

I will drive from Christchurch to Franz Josef, passing Authur Pass NP. Then from Franz Josef Glacier to Queenstown, from Queenstown to Milford Sound, then back to Queenstown and to Mount Cook. finally, from Mount Cook to Christchurch