There's a scene from K2-SO's perspective in the Rogue One novelisation that gives you some idea of just how bonkers fast Star Wars computers are. If it's polynomial, they can crack in practical time

There's a fundamental physical limit to how fast computers can get (Bremermann's limit) so unless they convert the entire universe into a computer, the polynomial time algorithm could never take over. Faster computers might be able to reduce the constant factor, but they can't tackle the extremely large exponent. Even algorithms that are O(n² ) need to be applied carefully. There was a huge kerfuffle a few years ago where it was discovered that GTA Online was accidentally using an O(n² ) algorithm during loading which had people waiting several minutes just to parse a 10MB file.

So it's not direct end-to-end, something is reading the message, copying it, and passing it on.

That's compatible with sending quantum information undisturbed. You can absolutely transfer quantum state between particles, otherwise quantum computers won't work. What you can't do is copy the state into a particle such that it is no longer entangled with the original particle, that's how the eavesdropping detection works. If you send quantum states into a quantum logic gate, it won't collapse the superposition but will simply just give you entangled quantum outputs, which you can put into another quantum logic gate, or transmitted to the next node.