Even at the National level, there are things to do with draft-type miniatures that may not have to do with their type or conformation, specifically.
Specifically, there IS "draft harness" in which you CAN style your horses after draft horses, right down to shoes (the ONLY class where you can have shoes except maybe chariot?) and braiding of the mane and tail. They even have tiny draft style harnesses with scotch collar housings (those pointy things that go over full collars). I've seen teams of six or eight to a miniature draft hitch wagon. It's the cutest darn thing.
Chariot, too, is always fun if you like to drive fast.
The hunter and jumper classes are not type specific, nor lots of the other performance classes. If they have leg action, drive them! There is also obstacle classes, both in hand and driven, which is not type specific.
There is plenty to do with a draft type horse, they just don't specify type in the halter classes.
Andrea