My gut instinct is you’d really need perfect incompressibility such that pushing one end of the rod would propagate the pressure wave to other end instantaneously. In other words, you have to make the speed of sound faster than the speed of light. I have no idea what the physical construction would look like. Maybe a string of singularities lined up and touching (hey, there’s your 0!) without tearing spacetime apart?

I think the math is fine, but a perfectly rigid rod cannot exist in our universe, so there's no paradox, even though it would let you send messages faster than light. It's like calculating how strong of a hydraulic press you'd need to compress matter into neutronium in your garage, or claiming that if a spaceship could travel at 2x light speed, then it could fly its way back out of a black hole.

There simply are no such things as rigid bodies in this universe. Everything is composed of particles that act on each other with forces, which take time to transmit the information that something is pushing on the other side.

My guess is that the Pauli exclusion principle would ultimately come into play, the same thing that keeps stuff from falling through other stuff. Electrons will shove other electrons out of the way, which isn't a lossless process or an instantaneous one. So a perfectly rigid rod would run afoul of special relativity if you pushed on one end.