Problem with chopper opamp(s)
The answer turns out to be simple. The output of the LTC1150 was not only being used as a reference but as the negative power supply of the ADC itself. It was sucking too much current from the opamp, and the resulting symptom was an offset voltage 1000x worse than it should have been. Unfortunately the LTC1150 data sheet does not indicate max current one can draw from it (other than short circuit current), or the degradation this causes.