Yes - agree with Greg on that. Fingerprints are almost impossible to remove.
You might be still able to get "finger cots" as well - like the fingers on a latex glove.
Ultrasonic cleaning should be fine - not tried it here, as you still need to spend a bit of time cleaning all the parts after this.
If the movement is visible it really needs a polish, wash, and chalky brush, so it is debatable if it is worth the ultra unless it is very dirty indeed. I don't.
If the movement is not visible I suppose you could omit polishing it - I do, but that's just me and not necessarily what everyone thinks.
Before you clean these (or any clock with pinned pillars), make sure there are no burrs on the pin holes so the top plate drops in freely. Makes reassembly easier. Pivots on these are glass-hard and will snap like carrots if you are not careful!