Keep up the good work!
We might be able to help you a little more when the development is more advanced.
Right now, I think you don't need 4 steppers for the z axis. First, it will be very hard to drive them synchronously.
Second, it will be costlier.
If you can place your mechanism at the center, you might only need one stepper. You might want to use low friction guide to prevent any flexion and friction. The printed bed and the printed plastic are pretty light and there is no need of speed on the Z axis, except for homing. You might also take advantage of using a gear ratio and lower the micro stepping. This is just my opinion.