I assume the shaft is not parallel with the bed and the likely problem is with the bearing close to the chuck. I also assume that you have a hole in the spindle shaft for a Morse taper. I also assume you have a Morse taper in the tail stock. The problem you have is to machine out the bearing block without a method of doing so.
I would start by lining up the tail stock dead nuts with the height of the back bearing in the head stock. Use the Morse tapers. Old broken Morse taper drill with the broken end made to join up with the end of a live center in a lathe tail stock. A collar could connect the two. Maybe you can come up with a better approach but this requires only making a special collar with a drill press. ID to fit the drill and the live center point short diameter The collar would have two set screws to engage the key way cut in the drill shaft, and a cutting tool can be mounted in the face of the collar. The bit is set to cut the OD matching the bearing. use the tail stock to move the cutter. Your using the lathe to machine itself. Your choice of bearing that can fit into the casting, roller, angular contact ball, or journal bearing. The basic same approach can be used to machine the rear bearing. If you use bearing you will have to keep the lubrication in and dirt out. Make or purchase shields.