As i don’t have a copy tool I have done similar with form tools for the balls if not too large or a ball turning attachment.
I would start with bar just larger than the diameter of the largest ball so you can hold the work without squashing your balls. Hold it in the lathe to do the ball on one end and the central one. Then saw it off from the parent bar, face to length and do the other ball.
Over to the mill to machine the flats on one side and also do the holes. Flip it over to do the other flats.
Back to the lathe holding the central ball in a collet or split bush and supporting the end with a female revolving ctr I would use the topslide to turn the taper. Flip it end for end and do the other taper.