To see where the weird 84mV on 1M range and 71mV on 10M range was coming from, I cut the 3IN+ pin of LM339 U115. After confirming no continuity between the stub of the pin entering the chip and the PCB pad, I powered on the unit.
At the PCB pad and at Bit 8 of the 4094B, I still see 84mV and 71mV in the two 4-wire resistance ranges. With a scope, I looked at the clock, data, strobe, and Bit 8 of the 4049B. The inputs all look sharp at least, but I couldn't confirm that they were correct. The Bit 8 output was a steady 71mV or 84mV, not bouncing around.
In this state, the meter actually reads reasonable values on 1M and lower 4-wire resistance ranges, but I suspect it's not functioning correctly on the 10M and 100M ranges.