I've climbed it in June and there was snow up to about your ankles just off the paths. However the snow up there is pretty varied, some years it's worse than others. There's tree cover up to a point, then the plains, which is where the snow is deepest but it's flat so pretty safe, then the peak is rocky. I think there are snow poles, but get in touch with the National Parks department if you're really concerned.

Considering I've climbed it with 80 other 13 and 14 year olds in a school group, in winter, you should be fine. It's supposed to be A LOT easier than the other six summits!