Quoted from Oldgoat:
If you were nearby, I'd let you borrow my veneer press (basically, I big, extremely strong plastic bag with a vacuum pump attached to it). Maybe you know some woodworkers locally and can ask about borrowing one? It is much more effective and efficient than using weights. Rather than water, I think I would use this product http://www.veneersupplies.com/products/Super-Soft-2-Veneer-Softener-Conditioner.html (this place also sells vacuum veneer presses). I've only used it on veneer, which is a fraction of the thickness of playfields but it is a much better product for wood than water.
A vacuum press is for applying even pressure on something you are veneering that is large or oddly shaped - it won't help with a bowed playfield.
Veneer softener is wonderful stuff for making thin material pliable, but it won't work on anything thicker than 1/8" in thickness.
A bit of moisture on the back, clamping and some heat to dry will snap the warp out of something like that - if not, then nothing will.