Take a page from the construction of the cake itself and put a post down the center of the Death Star to support the weight. Make it a cylinder and you get the best support/weight ratio. If you make the top of the post stick up from the stand you can attach a bolt to it, and the bottom of the cake plate can have a nut attached so they are screwed together. I suppose a hole in the top, and some regular pipe fittings from Home Depot would work as well. (Pipe of correct length, threaded on both ends with flange on the bottom, attach other flange to cake board.)
This is a nifty idea!