> doesn't that imply that the beam has compressed and is not rigid?

Yes it does. And in fact that is the case: The beam has compressed. A fully rigid beam is impossible.

What you're proposing is an effect mediated by physical forces. Specifically, on a fundamental, atomic level your beam is made up of atoms. And when you push it, you're bringing atoms closer together, that feel an electrostatic repulsion. And the beam moves.

So, in the best case, this method of communication is equivalent to communicating with electromagnetic waves. And they have a speed limit: the speed of light :)

So this wouldn't be faster than light communication.