So after giving up on this more than once, I spent a bit of time probing around the A/D board. The Elantec chip marked in red did not appear to be working correctly.
I also noted some ugly rework on the chip highlighted in blue.
I PM'd TiN to see if he ever saw a rework like this - he did not think this was a factory mod. (Thanks TiN!!!)
The Elantec chip is EOL and Keysight won't sell it, but I noted that there was a REV.C board on TiN's website that showed a replacement board that Agilent made for these two chips.
(No way was I going to source a chip from a Chinese seller on Ebay)
They are only $55 each so I ordered one and replaced both chips with the module. Note that this board (# 03458-66530) is "now available till gone", so there may not be many left...
Now fast conversions in AC mode is working, and the full self tests are passing!
No doubt that this will need a formal calibration.