I think you are correct
The lid will cup more than the "body" of the box because the material has been removed to keep it apart and it's thinner now and will dry faster. It would be great if you could penetrate the pores with an epoxy or other chemical like etheylene glycol. Maybe the wood turners will have an idea...?
The answer to your question will only be as detailed and specific as the question is detailed and specific. Good questions also include a sketch or a photo that illustrates your issue. (:< D)