Alright, progress! I had almost no time to look at it earlier so I set it up to log a 10V reference for a while (overnight) in the 10V range and then I set it up with a 20V (19.99997-ish actually) source all day in the 30V range. This tests the stability of the 100mV and 300mV ADC ranges as well as the x100 divider circuit. Screenshots attached, the meter is perfectly stable to 1 count or less. I did the same thing for a shorter period in the 1V and 3V ranges, no charts but the same result--perfectly stable.
Looking at the calibration constants it appeared to me that all of the DCV-related constants were way off while all the others seemed to reflect a reasonable amount of correction. Proceeding on the theory that this and the punctured calibration seal might indicate that someone has tried to calibrate it and failed badly, I ran it through just the DCV calibration steps. After that I scanned through and checked for linearity and offset on all the ranges and they were good. As I suspected from the design, the 30,000 count ranges are better than the 100,000 count ranges--the former were perfect while the latter were still good enough but not quite as perfect. There was a 3-count discrepancy at the negative limit of those ranges.
However, it appears that all of the other functions--ACV, DCA, ACA and Ohms are now in spec or at least very close to it. That's all I've got for now, more later this weekend.