Thinking about this...

If you sprung the motor sideways to pull on the belt and depended on the pulley flanges to keep the belt on the pulley, all your problems go away and you might actually get zero backlash from a cheap ball nut up to a useful vertical loading.

You can usually feel a bit of wobble on a cheap ball nut, not sideways wobble, rotational wobble 90 degrees out from the screw axis. The dust shields constrain it but the balls are slightly loose so you expect 20-50 microns of backlash after it beds in.

If you heaved the screw sideways using a belt tensioner, you would trap the balls between the screw and one side of the nut removing that backlash.

If the screw bends a bit, or the nut isn't concentric to the bottom bearing, the pulley will still turn concentrically. Meaning the motor never has to move appreciably sideways against it's spring during any single revolution and the load will not vary as it goes round.

This no top bearing idea could be a bonus