I don't think screws are going to last. The core under the glass is shot, thats why it flexes.
Without making a project out of it, instead of replacing the core, I would say to bond a new core on top. Its not a large area and you don't seem intent on a permanent "pretty" fix.
That could be as simple as silicone and a piece of pvc board or plywood. You need to spread the forces with a bond of the board to the existing glass.
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