If you drilled, tapped and faced your shaft blanks first, you might be able to mount them on your wood lathe with a mandrel of sorts. Then you can turn them by hand with a regular wood chisel. After taking a few light passes over a month or so to achieve a general tapered shape, you can take plunge cuts to approximate depth (leaving some room to sand to final size) at the joint, tip, and a couple points along the spine and barrel of the shaft. Once you connect those points you can sand everything flush with a sanding block to prevent unevenness. If you have any type of steady rest, use it. Chatter is a risk on a wood lathe using a hand chisel.
***These recommendations mostly come from my experience with turning bowls and pens, along with things I've seen/read on here over the years. A taper machine or setup on a metal lathe/tablesaw would be the ideal setup. I've never attempted the above methods, it's purely hypothetical and would be my plan if I was in your shoes and needed to accomplish the goals you've set with the equipment you have on hand. I'm not a cuemaker, but I have worked on cues in the past, so operating shaft taper machines, metal and wood lathes, etc isn't new to me. There are some smart people on here that may be able to offer better advice than I.
EDIT: Or you could send them out to be tapered as you mentioned... but why not simply just buy final sized shaft blanks to begin with?