The play is radial unfortunately. I am sure it is not going to be an easy or cheap fix.
I've made a new barrel (or should it be called the quill?) for a cylindrical grinder tailstock which was gear driven. It is a project, but possible. Here are the steps:
(1) Get the body honed cylindrical. Good hydraulic shops have suitable cylindrical hones and measuring equipment and don't charge a lot for this. Ask them to try and bias it upwards if possible, so that it ends up high not low. The biggest challenge for them is the pocket for the gear drive. The edges of the pocket need to be tapered and they need a hone that is long enough to fit completely through.
(2) Make a new barrel out of annealed steel that hardens well without much distortion. Turn it oversize, use a Morse taper reamer to make the socket.
(3) Put a new or good dead center in the taper socket, mount between centers (one female, one male), turn 50 microns oversize
(4) Mount in a mill, cut the gear rack. Measure carefully, you won't be able to check the fit until after step 7!
(5) Send out for hardening. (If you talk with the shop before picking the steel, they might be able to include it with a batch of similar parts from the same material, making it inexpensive). You want a place that will do the hardening in a vacuum, so there won't be a layer of oxide.
(6) When it comes back, use a couple of new Morse taper centers to lap the taper ID
(7) Mount the quill between centers as in (3) on a cylindrical grinder and bring it down to size. If you put a bit of taper on the rear 1cm, it will make it a lot easier to insert.
(8) Optionally, ID grind the Morse taper socket
(9) Rescrape tailstock (if too low, then also the headstock, see last sentence in (1)) to correct height and parallel in both axes
If you don't want to harden, use 4140 = 42CrMo4 steel.
Here's a few photos: