I agree the green is going to amplify the heat. You can use plywood as insulation beneath it and perhaps consider replacing it with white translucent. It won't change your dark and cozy nesting box set up. You might also consider lifting the roof a few inches to make vent space that goes all the way around (secured with hardware cloth of course).
Do they have access to that shady spot underneath during the day? I don't see a ramp or ladder but it looks like that's the intent. I would try to get them down there. My chicks have their original cardboard brooder box hanging out in the run area of their coop; they're not really big enough to figure out the ladder into the coop but they're not going to roost or nest at this age either so I figure it doesn't matter, they happily sleep in their brooder and roam the grass all day.