For myself, I use my barrel trimmer in a jacobs chuck on the lathe. Mine is the type where the central shaft is held in with an allen screw, and the shaft have a tendency to fall out on the pillar drill. If the shaft is horizontal, ie on the lathe, that doesn't happen.
As for your drill (jacobs) chuck, the tang makes no difference. It exists to locate in certain pillar drill tapers to stop it spinning, but it too is tapered, so should not be interfering with the quill your lathe. Giving the chuck a twist as you push it home in the taper will help, but if you are still having problems, retract the jaws so they are inside the chuck, then give the whole thing a gentle whack with a rubber mallet, which should hold it in place come what may.
Hope that helps.
