R&P is not really a good candicate for a Z axis due to the low resolution and low efficiency compared to a ballscrew.

For it to work and have anywhere near decent resolution then it will need gearing, obviously this ratio depends on the pinion diameter but from experience expect at least 6:1 to be even close to a ballscrew resolution.
By the time you have bought rack, pinion, 2 x pulleys, 1 x belt and then messed around with bearings and plates etc to make work then it's no longer cheap.
Then you have the CRAP factor.? . . . Z axis gets showerd with all the shit from the cutter and this gets into the Rack and can Jam and cause sticking and missed steps Etc.

A ballscrew from China complete with bearing blocks will workout about the same and bolt straight on with no messing and honestly R&P doesn't compare in performance and smoothness terms. R&P is notchy and clunky in comparison and on Z axis this is the last thing you want.

Stay clear is my Advise.!!