Thanks for everyone's help. After some deliberation I selected a compromise: I replaced just the middle hatchboard (the one with the 4" hole in it) by building a duplicate out of 3/8" acrylic. I cut two extra 2" bands of the acrylic for the top and bottom edges, to create the overlap joints with the upper and lower hatchboards and match their 3/4" thickness. I annealed all the edges with a propane torch then I welded the bands to the main piece with methylene chloride. This bonds the acrylic together quite permanently. Since it is 3/4" at the top and bottom it is very stiff and there is no chance of it flexing out of the frame (not to mention the overlap joints with the adjoining teak boards). It lets quite a bit of light in and allows you to see what is going on in the cockpit from the comfort of the galley. The acrylic is surprisingly heavy - this hatchboard weighs more than the teak ones despite being mostly half the thickness.
I was able to get scraps of acrylic from my friend's machine shop so the total investment was $6.40 for the methylene chloride and about three hours of my time.
I was able to witness the expansion of the acrylic with heat; when I annealed the edges the pieces curved; luckily they came back when they cooled.
