Lid cover
The big box lumber yards around here (Menard's) sell 4' X 8' sheets of white 1/8" & thicker plastic. Not the textured fiberglass stuff, but smooth finish, or very lightly textured. It cuts with a sharp utility knife or other tools. The 1/8" stuff might be too flexible.....but they carry it in other thicknesses. Given the depth of the depression, you could place a backer piece of foam at the approximate thickness of the inset and use the thin plastic sheet.
You could lay an oversized panel on your lid, adhered with VHB tape and use a router with trimming bit to trim to size. Or cut to size and glue with construction adhesive/contact cement, etc.
You got a box now, just modify it to improve or suit your purposes, or live with the problem.
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