If you're going to the expense of a CNC program to create the parts, you're better to 3D print with a resin printer. Can you CNC styrene, yes.....I've done it several times before. The problem you experience is plastic produces a great deal of heat when machined and often the removed plastic remelts into some sort of blob. To do it well, you need something like a blast of air to keep it cool and remove your waste. The 3D part would more than likely give you a better result with less frustration.