Is the pizza case kind of rigid, or is it floppy? I ask that because if it's floppy, the bungee cords will crush the box on the edges where they turn.

Maybe a prettier solution than the rigid wire/mesh platform is to fabricate something out of aluminum stock that you can drill some holes into. Run the members perpendicular to the rack, and bend them up at the ends of the pizza case.

An aside: What effect does the bumpy ride have on a pizza? Does the cheese "creep" off?

Also, in Philadelphia, I think the guys have special bikes manufactured by Bilenky to transport Pizzas. I've also heard of a guy at Domino's that put thumb loops onto a regular pizza bag and kinda just kept it on his handlebars. I'm not sure if that's feasible for your application.