A real angel would have no Enneagram, I suppose. Dudley, however, has many human traits. He feels romantic love for Julia. He takes a perverse pleasure in wrapping Henry around the axle. Even with the Professor, whose alcohol bottle Dudley magically replenishes, he likes confounding him and playing with his expectations. Dudley is an imp as a character archetype and, possibly, as a religious one. Henry has doubts, calling him a demon. I have doubts, too!
So, what Enneagram is the Mischief Maker? Dudley can ice skate beautifully, although I credit that with magic rather than a physical proclivity. Not a Body Type. He’s not particularly intellectual. His purpose on Earth, ultimately, is to save Henry’s soul. A different Enneagram, a Head Type, would argue with Henry and engage him in philosophy in order to guide him. Dudley, instead, goes right for the social levers.
In general, a Four is quite impish. It’s one of the things we love about them, that mischievous sense of finding buttons and pushing them, even their own. Dudley isn’t very self-reflective, though. It’s everybody else’s weaknesses he likes to manipulate.
I’m going to say that the writer purposely subverted the angel trope in order to create humor. It must’ve been funny in 1947, or it wouldn’t have become a beloved Christmas movie. In my lifetime, though, it’s gone from charming to drudgery. For me, The Bishop’s Wife is now unwatchable, and a lot of that centers around Dudley.