First, I have to wonder why anyone would want to put a 5C collet chuck on a Hardinge lathe with 2-3/16-10 threaded spindle that already takes 5C collets directly in the spindle.
Many lathes were built with 2-1/4-8 threaded spindle noses, including South Bend Heavy 10's. If you look at a catalog list of available threaded chuck back plates, that and 1-1/2-8 will be readily available, with few other options.
Here's what the lathe chuck looks like, standard 5c inside threaded outside. 5C collet chuck screws on the end. These are used for large diameter thin parts, not a standard 5c setup. This is on a CNC lathe. You might want to look them up.
I was with Larry on his comment. Why would you want a 5c adapter to fit a Hardinge spindle.
I think I see what you are concerned about. The spindle nose on your present adapter will not take the screw on collet closers made by Hardinge for step collets? If so whoever made that adapter screwed up or maybe they supply collet closers for step chucks that will fit the nonstandard thread. If they do supply closers to fit the standard step collets you will probably need just a few so and special charge would not be that great.
Step collet closers are on page 8 and 9 of this brochure:
If we are in fact talking about what Hardinge calls a step chuck closer, well...
I have a 1981 South Bend Lathe accessories catalogue which has "closers for step chucks" including for the 2 1/4 - 8 spindle nose of the heavy 10 and 13 inch South Bends. These look like the Hardinge step chuck closers. Also listed are "step chuck blanks."
I have a vague recollection of someone, possibly on this forum, reporting that South Bend step chucks did not fit the Hardinge closers. Let me stress that's a vague recollection.