The solution? Make a work holding device. This one started out as 1/2" 12L14 (free-machining steel) round stock, about 1" in length. In the back end, a large hole big enough to pass the entire screw (in this case an Allen head, properly called a Socket-Head Cap-Screw, or SHCS) was drilled most -- but not all -- of the way though, to make sort of a cup with the front end still closed.
Then in the front end, I drilled a much smaller hole -- and this is the clever bit -- tapped threads to match the threads of the screw. The length of the front hole is calculated to allow 5 full threads of engagement (4 threads would be sufficient, but I like to over-engineer). This produces a hollow tube. The screw is slipped into the back and threaded in so it sticks out the front, thus:
In essence, I gave my screw a really big head (hey, it happens to all of us now and again). Now I have a nice, big, 1/2" diameter 1" long item I can safely and securely chuck, and then machine whatever I want to on the end of the screw. (In the picture above, the dog point has already been completed.) Luckily, the lathe's normal direction of rotation tends to seat the screw threads more securely into the holder under tool pressure, so things don't go haywire when you start cutting. (Speaking of haywire, my first attempt at this sort of holder ended in an ugly mess because I made it out of aluminum. So, steel is definitely the material of choice for this job.)
When I'm done machining, I just stick the Allen key in the open back end to unscrew my screw from my holder.
In other news, I bought some 1144 steel (A.K.A. "Stressproof"). Not for any particular purpose -- just to have some on hand for as-yet unimagined future projects... and because I wanted to try it. 1144 is about 3 times stronger than mild steel, yet paradoxically, it machines wonderfully. Plus, it's a little cheaper than 12L14, so 1144 is now my second-favorite steel alloy. Still nothing compared to the sheer joy of machining brass, of course, with it's golden fountains of tiny chips; but sadly, brass is over 4x the cost of steel -- making it uneconomical for many applications. :(