After discussing with Koji, we looked at the aLIGO incarnation of this board. Interestingly, it too has a similar topology of 4 switchable gain stages with gains of 24, 12, 6 and 3dB. The main differences are that they use single Op27 ICs instead of the quad LT1125s, and also, they use a different combination of feedback resistors to realize the various gains. 

We considered upping the feedback resistance (R15, R143) on the 24dB gain stage of our boards from (1k, 66.5ohms) to (3k, 200ohms) as on the aLIGO boards - but this doesn't really help? Because KCL demands that the same current flow in R15 and R143, and so the output Vsat of the op amp and its max current driving capabilities in combination determine if the inverting input can follow the non inverting input?

As Hartmut points out in his note, he was able to access the full range of ADC voltages when the gain was set to 3dB, despite the fact that the LT1125 was still getting internally saturated. Operating with minimum 24dB whitening gain doesn't really solve the problem either because the problem just gets shifted to the next gain stage in the chain, and we still have saturation. I also don't have a feeling for how much differential voltage these LT1125s can sustain before they are damaged - I guess the planned THD check will reveal if they are okay or not.

It seems to me like the only way to truly fix this problem of one stage saturating and screwing up the others is to use single Op27s (or equivalent) in place of the quad LT1125s. The aLIGO design also has a series resistance to the non-inverting input - this can help prevent current overdraw from the previous stage (due to a lowered input impedance of the OpAmp - but I wonder how low this can go?).