You should be able to get a reasonable speed out the Z.

For comparison, I was tuning my Triac last night, and I was getting 2.2m/min with 500mm/sec2 acceleration on the Z, and that's with an old big round stepper driving a 5mm pitch screw with 12:30 gearing via a belt. Now I doubt your Z assembly is as heavy, and you'll be using a more modern motor.