As others have stated, you have an uncommon spindle nose. In the days when lathes with threaded spindle noses were in common use, backplates (or faceplates, driveplates,) were made in the shop where the lathe was in use. Cast iron backplate blanks were sold in shop supply stores as a regular feature.
Not knowing your experience level and how your lathe is equipped in terms of change gears (or quick change box ?) for cutting different pitch threads is the big question. If your lathe can be setup (either by combination of change gears, aka "loose change gears", or by quick change box) to cut 8 threads/inch, you can make a backplate in the lathe itself.
My own method is to turn a chunk of round stock to make a male threaded facsimile of the spindle nose. I call this a 'dummy gauge'. To cut the male threads, measuring the actual spindle threads using the three-wire method (known as 'measuring over the wires') is necessary. Asking a shop to make you a backplate is OK, but without precise measurements of the lathe's spindle nose, they are not going to be able to make a backplate that threads onto the spindle with a good close and accurate fit. There is an unthreaded portion of the spindle called the 'register', and that has to be measured with a micrometer. The backplate has to be bored to a very close fit on that register, aside from having a good close fit on the threads. Making the dummy gauge to match your spindle is step 1 if you are going to make a backplate yourself. If you are going to give the job out to another shop, then precise measurements are what's needed. No using a digital slide caliper... it's really a job for a micrometer and three-wire method.
Once you have the measurements, whether you make a backplate or pay someone to do it for you is your call. Backplate blanks for threaded spindle noses are still available but are pricey. I've made backplates out of steel, using a disc of steel plate and a chunk of round bar. My own 'druthers is to go for what is known as a "Complete Joint Penetration Weld" , or as close to it as possible, to tie the round bar hub into the plate disc. I turn bore the disc to a diameter a bit smaller than the finished diameter the round bar will be turned to for the hub. I then turn a section of the round bar to fit into that bore snugly. I put a heavy chamfer on both the end of the round bar and the mouth of the bore in the plate disc. This chamfer extends to almost the full thickness of the plate. I then weld the round bar hub into the disc, filling that chamfered area with multiple passes of weld. I also run a 3-pass fillet weld around the base of the round bar where it seats on the plate. I peen the weld with an air needle scaler to relax it, and "quarter" each pass to divide or try to equalize weld stresses. The result is a solid job, with the round bar and hub being as close to one piece as possible. I leave a 'land' at the base of the chamfer in the bore of the plate, and this land is maybe 1/8" wide. The land and shoulder align and seat the round bar, centered and square to the disc. I try to do a backwoods stress relieving on these sorts of weldments. Namely, I put the completed weldment into a wood fire or into the firebox of our coal fired boiler during heating season. Let it soak in the heat until red hot, let it stay there with the fire on low draft for maybe an hour or so. Then, into a bucket of dry coal ash to cool slowly. A day or two later, I clean up the weldment with wire wheel and needle scaler and can begin machining.
This relieves 'locked in stresses' from the heavy welding operation. I use 'stick' welding rather than MIG, and really 'burn in' the weld with E 6010 for the root passes, and E 7018 for the hot and cover passes. No wide weave of the weld, just stringer beads with maybe a little weave at most, to try to limit weld stresses and resulting distortion of the plate. This is why when an opportunity to buy a cast iron backplate blank came along, I jumped right on it.
Making a backplate lets you thread the backplate to a precise fit on your lathe spindle. Once made up, it is turned to diameters to fit the chuck(s) you want to use. It is best done on the lathe spindle itself, so the backplate is married to that lathe and runs nice and true. Without a chuck for the lathe, if a person had no other alternative, the blank could be moun ted on the faceplate for machining. A screw cutting lathe is an incredible machine tool in that it can make parts for itself, limited only by the ingenuity and skill of the person using that lathe.
