There's a bit of an issue with defining precisely what rigidity, reasonable height, and random orientation & angular momentum all mean. Also with the physics of perfectly sharp corners and edges. But my intuition says that the mechanical system of this polyhedral mass rolling around on a hard surface produces behavior which is so sensitively dependent on initial conditions that we could define almost any sort of continuous distribution on almost any small (but sufficiently energetic) subset of possible initial "die positions" in almost any approximately reasonable physics and still get the same set of probabilities.
My intuition also says it shouldn't be necessary to go through the pain and expense of actually performing any simulations to generate this set of weights. I have a really neat idea for treating this as a stochastic problem, but I'm stuck on scale invariance - I don't think it should matter if I've got a chunk of material a centimeter on a side or a meter on a side. But for any given distribution on initial conditions, it seems to matter a great deal.