Alternately you might consider a bolt-rope type fastening using one of a couple different extrusions available from places like Sailrite
You can sew some small stuff onto the edges of a flat square screen
on three sides so when you put the screen
on the hatch you slide the bolt rope
onto the middle of the three first, then do the parallel sides together.
If you make everything just right, so that you have to stretch the screen to put it in the extrusion, it will hold itself flat against the hatch without need for attaching the fourth side.
Likewise, you can use the same system to make hatch covers on the outside. Personally I think it presents a much cleaner appearance and has the benefit of largely eliminating the gaps that you have with snaps. Plus the snaps can stick and stubborn ones can break or pull through the fabric