Definitely sounds like the best option.If the bearings are good and you're happy with evrything else, just make a spacer washer the right thickness to take up the lash.
Put the nut on with a washer under it. make sure the hole is lined up. Fit the cross-slide up and use a dial gauge to measure the lash in the assembly by pushing and pulling on the cross slide. Measure the washer you used, add that thickness to the amount of lash you measured and then part off a new washer to that dimension. You should be able to now fit the nut on and tighten it up nicely so the locking pit will fit.
Maybe the movement on that threaded stub is intentional? Seems like a lot of effort to go to if they were just going to stick something undersized in the hole. It would have been very simple to make a tight push fit with no movement.