The thinking in the other post, that the mass increases as you move away from the center, in a manner that the two effects cancel out, intuitively seems like it should be feasible. Remember that the center of mass is just an abstraction, you need to take the full integral over all mass to get the force vector at each point. And if you're closer to more mass further away from the center, which a shape like the one described above should give you, it might work. But one would have to do the math to be sure.

Edit: come to think of it, maybe that effect would let you adjust the direction of the force, too. Thinking about center of mass can be treacherous with more complex shapes...