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.

Regardless of whether you are using water or a softener, the answer depends on how permeable the paint is. It could vary quite a bit based on the type of paint and how many coats (and how thick).

While I've never flattened a playfield, I have a good bit of experience with a veneer press. One sandwiches the playfield between two cauls (in this case, pieces of flat furniture grade plywood) and places the entire sandwich in the veneer press. The issue is one of cost. The cost of the veneer press, furniture grade plywood and softener is not insignificant. Hence my suggestion if you could find someone local that already had the materials you needed. Barring that, water and weights (or clamps) is essentially the same principle.

While I would be confident this could be addressed if it were solid wood, my concern is that a playfield is plywood. Where is the source of the pressure causing the bow? If it's in the bottom layer, the liquid will penetrate it. If a middle layer, liquid won't penetrate and to your point, it will snap back quickly. Wood can be a crazy thing.