I swapped over to a 3x longer cable (old 65 ft. Pasternak cable from ancient 40m days). The old one was 6m, the new one is 18.2 m. It was already coiled up so I put it into a tupperware box to shield it somewhat from the HVAC wind.
The noise went down nearly proportional to the length (after I recalibrated the DAQ channel for the ~3x higher phase->voltage gain). With this length, the peak-peak mixer range is 5.5 MHz, so still enough to go an FSR here.
I give credit to the low frequency improvement entirely to Tupperware for their excellent containers. The current noise limit is most likely the SR560.