That is a tacho you are looking at. They have no purpose nowadays, what with digital drives. You should try to remove it to see what the end of the rotor shaft looks like. A rotary encoder typically has a small (6mm or so) shaft and creates negligible torque drag, so mounting one in place of the tacho could be quite simple. Something like this Chinesium clone would be suitable https://www.ebay.com/itm/OMRON-Rotar...g/332386539771

One of those CNCdrive brushed servo drivers would be fine for a motor like that and the encoder would connect directly to it. https://www.cncdrive.com/DG4S_16035.html The bigger one is rated at 160V / 45A and although your motor is rated at 100A, it's a beast (47Nm!), so you can probably manage very nicely with half of that. The advantage of going this route would be minimal messing with the mechanical parts. You can probably just leave the encoders where they are for now unless you could hook them up with one of the Chinesium DRO displays just to frighten yourself.

This would be a good time to get a parts list / exploded / section view / manual for the machine. There's a good chance TOS would still have documentation, or you might find a user who could scan one for you. Then you can see how to dismantle it and how it works.

I would expect that the ballscrews will be fitted with double ballnuts, so you'd have some degree of adjustment for taking up some of the backlash. There's likely to be two ballnuts with a shim between them. You can fit additional shims to reduce backlash, possibly without having to even remove the ballscrew. Going to the extreme of fitting bigger ball bearings would likely require you to dismantle the machine, which is quite a large beast.