Finding the sources for INL in the ppm range is a really difficult task. It starts with the problem of reliably measuring INL in this range.
In the AZ mode I am not that sure the amplifier switching is not contributing to INL. At least in the old state some of the INL measured was due to switching artifacts. It might be worth repeating the INL test (compare to 3458 and turn over) with the modified amplifier.
One possible source of INL that came to my mind would be excessive ringing at the integrator input. After switching the S1024 current source, there will be some ringing of the OPs of the integrator. If the voltage at the integrator input would rise to above about 40 mV it would leave the linear range of the OP177 OP. Ideally there would be very little voltage there (maybe 10 mV alternating spikes that decay over a few µs). It is hard to tell how parasitic capacitance, decoupling and charge injection of the switches would influence things. Also the distance to C210 is rater large. So the circuit around the integrator might not be that ideal.
There could be a reason for this distance: heating the capacitor could change the effect of DA.
Another point to check with the scope would be some of the ground points, e.g.:
Both sides of U504 GND buffer, the center of R201, the ground at Q205, GND at U301 (HC74 driving switches), GND side of C207, U211 Pin3.
There will be likely some ringing when Q204/Q205 switch - too much of this could be a problem.
The substrate voltage for switches could also be interesting (e.g. U204 pin 7).
Another possible test would be to see how much warming / cooling of R200 would change the offset: e.g. warm R200 with a local heat source (e.g. resistor as a heater at around 500 mW) a little and than watch the voltage change in non AZ mode measuring a short, when the heat source is removed. This give an indication on how sensitive R200 is to temperature.
Similar other thermal effects are possible: heat source that depend on the input voltage should be R200, Q200, U206, U207, R219, R220. Possible sensitive parts would be the ref. scaling (R232, R234) , R200 , R201 and also C206. For C206 a change in capacitance would not be a problem, but a speed up of the DA can likely have an effect in AZ mode. Still I doubt the heating would be enough to increase the temperature by some 5-10 K needed to get into the ppm range.