My only suggestion would be to chuck it between centers and turn a tenon to fit your chuck jaws. You will still need to keep tailstock support. With my chuck and the 45mm spigot jaws it states a working length up to six inches for unsupported work(and that is with one inch or wood buried in the jaws). There are a lot of forces at work even six inches out.
Hopefully you could turn the basic shape with the pointed end down to maybe 1/8” diameter. Then shape the large end down then to 1/4, 3/8 or your preferred diameter. Carefully finish off the point, then saw off the other end and sand the nub.
Good luck with them.
They who can give up essential liberty to obtain a little temporary safety, deserve neither liberty nor safety. Benjamin Franklin