The slope-amplifier usually has additional back to back diodes in the feedback, so the OP is not driven so hard into saturation and also the comparator sees only a smaller signal. In addition a cap to reduce the BW may be good. A single slope amplifier for both comparators can be OK if the 2 trigger levels are similar (e.g. within some 400 mV after the gain). The less critical comparator could also use to signal directly from the integrator - usually only one comparator is used for the start / stop of integration, though in theory both could be used with averaging for reduced noise.
The resistor values are a balance between the noise and the current drawn: there could be quite a bit of current with a large input from the integrator.
I don't think it is needed here, as the quatization noise is more limiting, but as an alternative solutuion one could replace the low resistor with 2 JFETs, to give a low resistance at low current and than limit the current to some 1-3 mA. The exact gain of the amplifier is not critical, only that it is reasonable stable for one convesion. I would consider the main advantage in a reduced (controllable) bandwidth, not so much in having a lower noise density to start with.
With reusing the µC code in a new board there could be a copy-right problem. So there could be a catch with the stand-alone idea.
Ideally it would be a new software and just get the info on the interfaces / messages. Still not sure if this would worth the effort. The ADC is still not very low noise, slow (or limited resolution) and quite some effort. I may be a bit biased here, as I have my own ADC design with less effort and noise. However it is made for an AZ mode in the front end and thus not easy to use in the 8081 environment.
The jitter is slightly less critical with the 200 Hz forcing signal, than with the clock to the HC175 flip-flops. If the 10 MHz µC clock would be used, one could also use a µC timer to create the 200 (or 400 Hz) signal - so no need for another divider chain, just switching of the clocks.
As far as I see it the CS2000 should normally run with 3.3 V supply - though 5 V supply is still in the absolute maximum range. Lower supply would reduce the supply current and spikes. Not many parts actually need the full 5 V (e.g. the RS232 and external control).
The LT5400 resistor array should be OK from the noise. Normally the higher tolerance version should be OK too. It is not so impoartant that the 2 are really accurate, but they should be very stable over the time of a experiment (e.g. a few hours). Drift of the resistors results in offset drift of the ADC. 1ppm ratio dirft would correspond to some 16 µV ADC drift. So there is a reason one wants resistor excess noise levels very low (e.g. better -50 dBi).
Noise of the amplifier for the inversion comes to the result with a gain of 1.6 (or whatever the resistor ratio at the integrator).
The capacitor C1 looks quite large to me. I think one should get away (and get a better settling after jumps) with a little less capacitance. I see no real need for a PP cap here. A smaller size MKS type should be sufficient. Similar C2 should be OK with MKS and less size to reduce the parasitic capacitance.
The max313 not switching correctly and thus a floating resistor could be an INL issue, though this is hard to tell. If the 2 compararator levels are close togehter, the high Z phase would be only quite short and not too variable in length. Some 50 Hz hum on the 5 V supply would be less of an issue, but if the state or switching of the HC175 or the 200 Hz signal has an effect on the 5 V supply and this effects the 10 MHz clock at the CS2000, this could be an issue.
My main suspect for the square law INL part is an effect of the charge injection of the max312 switches to the +-10 V reference signals. With rather large 3.3 µF caps, the recovery of the reference signal can take quite some time. So while the large caps help to keep the spikes small, they also extend the time. It may be worth testing smaller caps there, so that the reference signal would recover in less than some 100 µs. Short pulses would more like cause an offset only, not INL. INL is possible if the recovery takes longer than to the next time the reference is switched. It may be good to have seprate buffers for the forcing signal and the main reference signals to the other half of the max312. The extra buffers are quite some effort.
For linearity testing calibration of the 5440 calibrator should not be very important.