Unsurprisingly, the mini-lathe lacks enough stiffness to apply enough force to hold a disk in place while turning its rim:
The old South Bend lathe had mojo, but those days are gone.
So drill and tap that fixture for an M3 screw, then stick some coarse sandpaper to it:
Snug the screw (a Torx T9 from the Small Drawer o’ Random M3 Screws) down on a rough-cut disk:
Sissy cuts remain the order of the day, but the screw applies plenty of clamping force and doesn’t require the hulking live center.