I forgot about the "option" of using a draw bolt . :redface:
When you say the "chuck spins off" what exactly do you mean ? Do you mean the chuck comes off the short tapered part of the arbor that sticks out of the headstock ? . If that is the case, a draw bolt won't help. . A draw bolt is meant to hold the long tapered part of the arbor in the (whatever it is inserted into - in this case the headstock).
Now, getting a MT1 tapered arbor which is threaded for a draw bolt ..... hmmm .... don't remember ever seeing one. MT2, yes.
But I think, as Ernie said, you would be better off buying a bottle-stopper "chuck" and screwing it onto the headstock spindle thread (using an adapter if necessary).
https://www.pennstateind.com/store/PK-BS1-MJ.html
But if you notice, they sell you a drill bit at the same time as part of that purchase. The drill bit is sized to drill your blank to accept this insert:
https://www.pennstateind.com/store/BSERT.html
But the insert is not really necessary. Just drill the right size hole to screw the B.S. "chuck" directly into your blank.
But the first thing you have to do is find out the thread size on the nose of your lathe spindle. It's probably 3/4-16 which means you will need an adaptor in order to screw the 1-8 "chuck" on.