That three-week window is the holy grail. The base is deep, the crowds from the Australian school holidays (early Jan) have left, and the Chinese New Year (which moves) hasn't always hit yet.
If you want the path of least resistance: You won't get the absolute coldest temps, but you will get the deepest, most forgiving powder of the season.
But timing is everything. Book too early and you might find dust on crust. Book too late and you might be skiing mashed potatoes. So, what is the actual best month to ski in Japan?