I'm still thinking nylon or other plastic is the way to go. The clip bearings Unworthy1 linked to, above, seem not quite right to me, but it all depends what's out there. Ideal, in my mind, would be a pair of plastic rings that nest together on a conical surface and, together, form a torus (donut) shape that fits the existing cup and cone. I rather doubt anyone makes exactly the thing I imagine, but I know there are lots of manufacturers making lots of plastic bearings. It's just a matter of finding the best one for the job.

A perfect fit is not necessary; extra space can be filled with epoxy or whatever.