Hey, you guys (Duncan and Pete) :- PSI has a product they call a "Between Centers Mandrel" ....... where ya been ?
Check it out ....
https://www.pennstateind.com/store/PKMBCM2.html
I don't have one .... wouldn't know how to use it .... but there it is.
BTW ..... I turn between centers all the time !!! . . LOL !!
... using the well-known and generally recommended tailstock live center and headstock dead center
both of which have conical ends ( 60 degrees, I believe ).