I have never shot through mesh either with shotgun for safety.
However, I have made my own mesh cover and just have the barrel of the gun extending out.
Its really easy to to. Buy a big roll of camo mesh and wooden clothes pins. I unzip my doghouse blind all the way and pin the mesh to the roof and sides, so it hangs down, and then pin in to the unzipped blind window (the big window not the small window with the mesh). The only spot not pinned is the bottom. This makes it easy to just put the barrel out, and you are still totally concealed.
I never like those small mesh windows and certainly not the velcro ones.