If the foam is melting, it's because you're going too slow, not too fast. Or you need a different type of cutter. With the right router bit, you can cut foam at 1000 ipm or more. Melting foam or plastics, or burning wood, is from too slow feedrates and/ or too fast spindle speed.
As for the power supply, it will probably be fine. Doing the math it comes out to around 11a needed. Most likely you'll need less.