A central supporting 316 stainless spike like the little plastic table they put in take aways?
Woodworking
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just keep cross grain situations in mind.
I think you're on the right track with reinforcements. If it's moisture from the pizza, and I'm inclined to agree it is, then sealing it with something more permanent might be in order. You can mix together two part epoxy, then thin it with rubbing alcohol and brush it on.
I wonder if, as an alternative, you could use a thin piece of aluminum the same size as the lid to distribute heat, provide structure, and act as a moisture barrier.
Your plan is sound. Just make sure whatever you use as a brace underneath doesn't naturally warp in the same direction