> This tetrahedron, which is mostly hollow and has a carefully calibrated center of mass
Uniform density isn't an issue for rigid bodies.
If you make sure the center of mass is in the same place, it will behave the same way.
If the constraints are that an object has to be of uniform density, convex, and not containing any voids, then you cannot choose where its centre of mass will be, other than by changing it shape.
If the constraints are that an object has to be of uniform density, convex, and not containing any voids, then you cannot choose where its centre of mass will be, other than by changing it shape.