If we knew more about the part and work needed that would help. The 1-degree part's Od taper would be a self-locking taper.
It may be possible to use a 3/4" round stock and with that held in any chuck for machining/makimg a tube-type taper ID holder
Stock 3/4" x 3 (or what the OP figures the length after reading this post.
Skim the OD of the tube type holder to run true (perhaps .748), then drill it (well centered), then ream it with the ops taper reamer, Parting-tool part it off at a length suitable to allow 1/8" of the part big end sticking out.
Now cut off the small end of the tube with having a taper Id, so it allows 1/8 " part's small end to stick out (for knock out).
This tube holder would be set in a 3/4" collet so the part's small end is pointing out toward the tail when in use. This taper tube holder would have no slits, just a straight Od with a taper ID.
The part would be tapped into the tapered ID tube for a taper fit holding...and it(part in tube) put into a 3/4" collet with the small end facing out and the machining would be performed.
When done the tube holding the part would be taken out of the collet and the small part end tapped on a brass block to release the (taper fit holding)part from the tube.
Some fineness (Trial and try) to obtain the correct amount of tap in, so the tap out will work.
* if the Op thinks the machining of the part would be too much for the holding he could make the taper ID tube holder by having a stop-screw to at the big ID end of the taper tube holder.
Sich a holder tube could also be set on the reamer(if the reamer has centers)so the tube OD could be run/turned between centers for a near-zero accuracy.
Such a holder could be made from a 1" stock, and set between centers to turn down to 3/4".
it could be made of 1" stock if the Op's lathe can hold a 1" collet.
This tubetype holder would use/go into a standard straight collet.