Use an expanding mandrel, but you probably can't make one). What you could make is what you would describe as conical shaft collars. It grips, is self-centering, and handles variation in bobbin hole diameter.
You can can use plastic, wood or metal, but wood cones seem easiest to just buy off the shelf.
The ideal setup is to have an interrupted shaft with one cone fixed to each shaft. One shaft is driven while the other idles and can slide to open up or close the "jaws". The idea is to clamp the bobbin between the cones via the bobbin holes. Makes bobbin installation/removal quick and easy.
If your shaft is predetermined and cannot be split or you cannot accommodate a double support for each shaft so it stays straight as a cantilever, a bit less ideal is use a single shaft but add in holes so threaded rod can be used to pull the cones together on the shaft. Lop the tip off the cone so it large particularly large circular faces are present, the drill the center shaft hole. With the extra circular area in the faces drill two, three, or four symetrically distributed holes around the center shaft hole to insert threaded rod through which is only used to pull the the cones together to clamp (if the shaft is driven). Then you also need a method to get the cones to rotate with the shaft while also allowing sliding to clamp. Keyway or square is best but I suppose you could have a flanged shaft collar to lockdown the cone to the shaft after clamping.
I only recommend this approach or the first one. Everything that follows is basically why other approaches suck.
Or you can one continuous shaft with and the cones thread onto the shaft to clamp the bobbin. Optimal results here require only the center area to be threaded and the rest to have a diameter equal or smaller than the minor diameter of the thread, but you probably can't pull this off. Otherwise you end up needing to screw the cone down the entire shaft, and if the cones are not metal, threads durability is an issue, especially against the threads on the metal shaft. Threading is also a pain. More work for worse results compared to the previous approaches.
Least ideal is to actually use them as conical shaft collars. Ideally you want the motion of tightening to also clamp on the bobbin. This is why setscrews are not desirable since the direction pressure required isn't inherent with setscrews.
If setscrews, you need to put a flat on the slope of the cone so you can start a drill to tap a setscrew. Or turn down the outer end to be cylindrical so you don't need to try and put a flat on an angle. More complicated work than the other approaches with by far the least desirable result.
Or, easier, is to bolt a flanged shaft collar to the cone.
The shaft collar idea won't work. Though it is possible to use a helical reamer to ream in the presence of a discontinuity (i.e. set screw hole, keyway or split) in the case of a split collar you need to tighten it while indicating it to make sure the bore is circular first. But since the bore isn't circular to begin with you can't indicate it to center to begin indicating for circularity.
And then reaming is best done on a lathe, or at worst by hand. Drill press is asking for trouble.
Also, look at model airplane propeller balancers. You may find conical components you need.