Thank you to everyone for the replies.
One thing I had not thought of is that the unthreaded back of the face plate hub may be smaller diameter than the register on the spindle (and I learned a new name for a spindle part).
Another is that the face plate may not be made by South Bend and even then may not have been made for my lathe. I seem to think that I did not buy it with my lathe.
My lathe Unit Code No. B 100 NK
Serial No A 9775NCR10
The face plate bears the number B 100 N. This is different from the face plate numbers in my South Bend Catalog.
First priority is some measuring.