Thanks for the suggestion, Gordon.
My drill chuck is not threaded. It came with a shaft that I had to jam into the chuck itself, and the other end of the shaft is the #2 MT. Much like this one
I'm not sure it's got a threaded hole for something like a drawbar. I'm also not sure that my lathe will accommodate one, either. Not in the headstock, anyhow.
My main problem was being in a hurry!
I didn't want the extra steps involved in removing the dimple from the live center in the tailstock. However, those few extra steps really saved me a lot of trouble by stabilizing the workpiece in the first place.
As a side benefit, if I use the tailstock when I shape the piece, when I'm ready to pull the tailstock away to finish the end, it turns out the drill chuck stays in place.