Ouch! Poor pizza! Looks like longer bungees might be in order so that they're not under quite as much tension when holding a pie.

I do the plywood thing for picking up doughnuts. Two pieces of thin plywood with notches cut in the center of each side (so that the bungees stay in the notch. I put one below and one above the doughnut box. I don't bother mounting them to the rack.

Single piece of plywood below the pie would likely work for you.