My grand kids painted some things I made them from foam board, they warped too. I don't remember what kind of paint they used, but it would have been for kids, they were small at the time. I like Rich's idea of covering it with Plastruct, that sounds like a good idea to me.
There is something called Gator foam or Gator board that I believe is supposed to be paintable without warping, but I've not yet tried any of it. For that matter I don't recall ever seeing any in person either? Saw an add for it somewhere a while back, but I'm sorry I don't remember where it was. Google search maybe, if you are interested.