Seem to be a rather tough challenge to get down the noise.
One thing might be doing a real noise measurement, to know better what noise is there. It looks like the ADC is continuously running and different resolutions / integration times are just a question of reading at different times or doing averaging in software, which would result in the same. So one would get the most information from taking data at a relatively high rate, but still not missing any conversions or too much resolution, and than do FFT on them.
C201 might still be an issue, as the original specified polypropylene type, not MKS. Through I would expect more like minor non-linearity from this.
If we are at caps, C212 might be critical too, if it shows leakage.
With the ICL7650 for IC201 there might still be a problem: this OP is known to show higher than specified noise in several cases. Though the noise from this OP should give something like 2 µV_pp in the 0-10 Hz range only. So for a 400 ms integration time this should be really low, even if several time the specified noise.
IC203/204 might also want local decoupling. Though I doubt they are the noise source, changing then could be relatively easy / cheap. There was not much jitter at the output of IC205, so I doubt this chip is at fault.
Another point might be the reference section. The ADC relies on the positive and negative 10 V refs to be connected. So noise in the ratio will show up as noise of the ADC, even if measuring 0. So IC306 is critical in this way though at nominal specs it only contributes something like 0.2 µV_pp for 0.1-10 Hz. The forcing waveform might give some dynamic load to the references and thus could give minute ringing that might also contribute.
Another point worth looking at might be the cross over of the two OPs in the integrator, I don't know how accurate the design is (likely done before spice simulation became popular). It might be worth simulating that part and see if it might react sensitive to tolerances in the caps or amplify noise from some bands.
p.s.: I did a short similation: the cross over and tolerances are not critical at all, so not problem here.