i have the smaller HF metal lathe.
i was out playing with it today and realized that my live center is a little large.
Mine was hardened as well, but tnat means 59-62 Rockwell C scale. A good C-6 czrbide will cut it nicely.
If that little tool you have pictured came in a set, use the left handed one (the one that would turn towards the tail stock), and use the set up just like you have pictured. Jam the center into the part, and feed by hand. Put a line on the center where you need the clearance to be, and work slowly.