[Wooffy]

Not sure on the diameter of the button hole fastening you are seeking to accommodate, but i do know Styrofoam (through elements of my work). Depending on the thickness of your foam, you should be able to get a very neat hole easily cut through with a hand held apple corer. As to how to fix the badge itself - I would suggest thread / fishing line on the shank of the fastener which could be tied to something stretching across the rear of the hole - this could be recessed into the rear of the Styrofoam if need be. Pulling it tight against this retaining pin/bar/piece should hold it securely and not damage the badge. For a professional finish, I would only just cut a thin slot through the felt and slot the badge into it, the badge will be held tight against the felt by the tension in the thread / fishing line.