I can’t see that it would be terribly difficult to make in a typical hobby lathe such as Myford or minilathe or Boxford etc.
But you would need a good precision adjustable ball turning tool. ISTR that the special edition of MEW a few years ago with a collection of “best projects” from the previous 25 years or some such included drawings for such a thing. Not terribly difficult to make. Or perhaps a boring head converted into an “up and over” ball turning tool ala GH Thomas would work. Certainly his purpose made tool would.
With that taken care of, the rest is fairly plain sailing. I would turn the ball on the end of the 3mm stock first, with a groove to depth where the end of the tapered section will eventually come. This would allow the ball turning to be done with minimal overhang, essential on such a long slender job.
Then slide the stock out of your chuck (used with brass shim around the job for protection) or ER collets etc to allow turning of the main body. Taper can be set correctly by offsetting the topslide, using a dial indicator to measure taper per 25mm etc. Test it first on a piece of scrap. If needed, the end could be supported by a very small revolving reverse centre cupping the ball end — another special tool to be made.
The the larger diameter collar and the parallel end section can next be turned to size.
But before parting off, grip the job by the remaining stock in collets or chuck in the mill and mill that small flat. Or hold it in the toolpost in suitable holder and mill it in the lathe. Then part it off to suitable length.
I can see why a commercial shop would say it is too small for them. Not because their large lathe could not do the job but because they could not charge enough to make such a fiddly small job worthwhile. Especially to do four of them as “one-offs”. And they probably would not have the suitable small precision ball turning tool or small reverse centre.
So I think David George’s method of turning the tapered section and soldering a ball on the end is probably the most practical solution.