Perhaps you should store it in your car-shoe-caddy...
Downside: all the other parents might hate you.
Have a tooth stamp? You're even further ahead of the game.
This totally works until they move out.
This also works for childless couples who just want to eat together.
And never have to do the awkward-reach-under-the-seat maneuver again.
They're crawling around on the floor anyway, right? They might as well help out...