Koji suggested looking at the output of the AS55 demod board on a fast oscilloscope to look for differences in the two channel outputs (if there is some high-frequency oscillations, for example, we could miss this information in the SR785 spectra). Besides, I was only looking at spectra out to a few kHz on the SR785. I grabbed this data with a 300MHz BW Tektronix oscilloscope (battery mode) today. Input impedance of both channels were set to 1Mohm, and the measurement was made with the RFPD input terminated, output of the daughter board is what is measured. The vertical scaling of the channels was set to the minimum allowed, 1mV/div.

Attachment #1 shows that there is indeed a visible difference between the two channels - the (noisier) I channel has a much larger DC offset of ~5mV compared to the Q channel (I tried switching channels on the O'scope and the larger DC offset remained on the I channel, so seems real). There is also some kind of oscillation going on in the I channel, although the frequency is pretty low, with the peaks spaced ~50us apart. Indeed, in the ASD of the acquired data, the excess power in the I channel at 20kHz and higher harmonics are evident (see Attachment #2). Anyway all of this points to something being anomalous on the daughter board I channel signal path - I will pull it out and monitor the outputs at various points along the signal path with the fast scope to see if I can narrow down what's going on where.