It seems to me that it is not possible to move an entire Gradebook Category (with its subcategories) into a different Parent Category. Is that correct?
You can move a category with all it's sub categories. You just have to use the movement icon in gradebook setup:
Then click your destination (at whatever level of russian-doll style category nesting you prefer):
You just can't do it in the edit settings for the category - that's where people get confused on this one.