Woodworking

I'm making a box to store a pizza. The box itself is pretty straight forward, cherry wood, box joints and basically a groove in the bottom with a piece of plywood (cherry plywood) to act as the bottom. I'm pretty happy with how the bottom went together but for the top I'm having an issue....

So for the top I have a 3/8ths piece of roughly 10x10 with basically a dado around the edge so it insets slightly into the box. The problem I'm having is that when a pizza goes in, it bows a bit. I'm guessing it's the steam from the pizza but so far I've been able to put something heavy on it and it comes out but I want to prevent it in the future.

I'm thinking a couple strips on the underside (basically across to U that forms) might help (with some glue and a couple of screws). I've also "sealed" it with a food grade oil but I'm looking for thoughts if this will or won't help and any possible alternatives.