I already started to modify the axis by using the "floating" setup, where the lead screws are detached from the X axis. I used this setup:
Ohmeye z axis isolation
This wasn't enough and I ended up reengineering the whole Z axis setup. The last part was a support for the lead screws, so they were no longer hanging from the stepper motors. The idea is that we put bearings below the lead screws so they are supported, can rotate freely and have no side to side movement.
First we printed a simple part support part.
It holds the lower end of the bearing.
Next drill some holes to bolt it to the printer.
This now bolts to the bracket from the printer.
The whole part can now be mounted on the printer.
As the leadscrews will end up a bit higher, I changed the configuration of the motors, so they are now moved up above the mounting bar.
Now the to part of the bearing can be put in place.
Next we milled a holder on the lathe. This slots into the hole in the bearing and has a hole on top where the lead screw slides into.
The flexible motor couplings are replaced with the ones below.
Now the new (straight) leadscrews can be put in place.
It all seems to work very well. No more side to side movements on the Z axis, movement is now very smooth.