Furnace filter fabric would block some light without blocking all the airflow.
That would work without the whirly bird on top if you can open enough of the open side - basically take off the wall. It becomes a three-sided shed. It is called a Woods' Coop because Price T Woods spread the plan in the 1920's in the poultry news journal he edited and in books he wrote.
It works best if the open side is to the south but any direction will work.
Pool noodles are a quick, easy, cheap way to block light coming in the eave openings of the three closed sides. That would make it easy to open them if you wanted to return the coop to the current design.
Painting the ceiling a dark color would help a little especially if you can add texture too.
I tried putting rockwool insulation in a well house ceiling. It was truely awful to work with - heavy, itchy (and I'm quite sure it is important to not get it on your skin or in your lungs), falls apart easily.
Other suggestions...
If you build baffles, angle them rather than put them parallel or perpendicular to your coop. That will help deflect the sound waves. Maybe deflect them up.
If you build baffles, you can gain a little more sound resistance by off setting the supports and fasteners. I don't remember what doing that is called but if you look up sound proofing in drywall for bedroom walls it should come up.