Hello I work for a company near Chicago that used to represent Hardinge for over 70 years untill they dropped us this year. I feel bad for all Hardinge customers b/c the new reps dont understand the equipment/controls well.
Seeing that you have replaced that expensive BZ sensor more then once takes that out of the question. Usually if that sensor is bad or "flakey" threading and syncing the sub to main has issues.
That sensor goes to a junction box underneath the sub's collet closer. From that junction box runs a cable to the sub's Amp/drive. It could be possible that, that cable could be the suspect.
The Amp/drive can also be the problem. If you look on each of the drives there is a label on the top cover. (covering the DC buss jumpers) If there is 2 of the same drives in your machine, you can swap the 2 and see if it follows the axis or drive. b/c its possible the drive is doing it.
Also on the control's PCB unit. "the mother board" there is a board on there that interfaces the drive to the control. Could be possible that, that board has the issue.
If you have no luck I would suggest to get a Fanuc tech in your building.
I would also check the Sub's belt condition/tension
I would also check the Sub's brake to make sure it isnt rubbing on the drum or mis-aligned.