Mounting an AXA QCTP on a 9x20 lathe

[metric_taper]

Looks like you have those infamous step pulleys. I can't tell what kind of expanding mandrel you have there. I've seen the type that has a tapered axle with centers on each end, and then a sleeve that has the inverse of that taper that was cut into sections that lets it expand on the mandrel.
I have used that on my lathe the issue I ran into is I could not drive enough torque into the work. And if I recall it's hardened steel, the drive dog set screw could not bite in, as it was slipping also.