Should I spend my time equally in North and South Island? I only have 12 days.

If you have less than 16 or 17 days, our strong recommendation is to spend it all in the South. The South Island is far more rewarding for a motorcyclist that the North Island. The South is far more spectacular and diverse from a scenery perspective, better motorcycling roads, much lower traffic density ( the South Island is the size of England but has only 700,000 people) plus it has better weather in Summer. You can ride for less than 30 minutes and see the scenery change from lush coastal rainforest as in Oregon, to an alpine pass a la Switzeralnd, to mountainous but barren moonscapes as found in Northern Israel. If we were asked to rank the 18 best things to see and do in New Zealand by a motorcyclist, the first 16 would be in the South Island…
Which is why our 14 day guided tours only cover the South Island. Only our 21 day tours head up into the North Island (for 7 days of the 21)