I hope I am reading the problem right.
My initial thought is, to clock the internal taper of the spindle to work out exactly what the taper is - decimal or inch - i.e, x fractional inch taper per y inch linear - or metric equivalent. Then, see if there is some way to set up the adapter by some means - mandrel of some sort and get it running true via a 4-jaw chuck... this is the awkward bit! Not sure yet quite what I'd do without the adapter actually in hand!
If that can work then the adapter can be clocked to see what the taper discrepancy is exactly, then set up for the correct taper and grind externally or even cut with a carbide insert even if hardened. Now, if this could improve things the next step could be to lap the adapter and spindle gradually together until the fit is correct - which would mean then having to always use that spindle taper/adapter combination for good. A good 1/10s indicator would be useful.
One other thought - if internal taper of spindle could be measured well enough - use that info to turn a new dummy adapter out of ordinary CRS and try and get that right for fit. If that worked then drill and ream for end to take a Morse dead center. It could serve even if not hardened.
Sorry long winded but sort of thinking aloud right now