Hello.
Yes : ducks do poop a lot while they sleep.
Nevertheless : I would really recommend you to NEVER use hay in your duck house.
(I don't know where you have heard it was good to use...? I was pretty sure it was a known fact hay is not considered safe to use for a duck house?)
...Personally, I just use straw for my duck's bedding.
And I am happy with that - only issue I am having being the fact straw is becoming rare where I live (apparently partly due to the fact people built houses on "former" agricultural lands), and so, harder and harder to procure...
I don't know about using shavings, since I have no personal experience with these... (Sorry.)
(...Also, you would do good by waiting for your - so cute - ducklings to be sexually mature before putting them with your drake : indeed, you don't want him to kill your ducklings - be it by raping them to death, drowning them, or whatever he could do to kill them...)