I have some great pictures on my laptop, which is at home I’ll try to upload them and post a link. In the mean time I’ll try to describe a set up I saw at the Seattle boat show.
They had companionway doors (barn style) that had glass on the outside and screens inside. There was trim all the way around but when you’d look at the top end of the door, there was a handle that when pulled up was attached to the glass. This looked like just a piece of trim. It sat in a channel, just like your hatch
Another description is they built the frames of each door out is say ˝ stock. Then they placed screens on the inside of that and then finished the inside with trim. The outside had finish trim placed over the face frames but then overlapped the open area for the glass a ˝” or so. There would be a small groove routed into the frame for the glass to slide in from the top.
Like they say a picture is worth a thousand words.
There was also a thin groove on the inside of the overhead slider rails that had a screen too. This retracted into the cabin top. You can open the slider, but then close a screen.
It was a really nice looking set up.