I used a Eurocard extension board to peek at the inputs and outputs of each of the gain-ladder AD829s on input B of the CM board in the +31dB configuration with the input terminated. (i.e with the following stages active in this order: +16dB, +8dB, +4dB, +2dB, +1dB).
The voltages I observed imply that the +8dB stage has an input voltage offset of -2mV, whereas all the other positive gain stages show around +-0.5mV. This could explain the shift observed at the +15->+16 transition. (However, since both input channels show a jump here, maybe its something more systemic about the board...)
In any case, it should be simple enough to swap out a new AD829 in place of U9B and see if it improves things, before getting too deep into the muck. (In principle, the AD829 has offset nulling pins, but I'm not sure how to do it in a non-hacky way since the board doesn't have any pads for it.)