You need to at least set screw the pulley. Grind a flat on the shaft and it should be fine.
With only one bearing you will have backlash but if the head slides easy it may keep preload on the nut and screw.
The stock screw will have lots of friction so you will need a large stepper and a good amount of reduction to lift the head with a stock screw.
A 13T to 36T and a 500 Oz or larger stepper might work but I haven't seen anyone do it yet.