Whilst constructing a new flange to replace the old one, we did some tests again to compare the performance of new vs old flange.
See attached images:
1. Old flange, in the lab, no grounding attempts made, biased, n-sides.
2. Old flange, lab, ground plate connected to flange, biased n-sides.
This comparison shows that for the old flange, connecting the ground plane to the flange improved the 50Hz noise. This was also true in the ESR, so all following screenshots had this 
ground connection.

3. Old flange, in the ESR, grounded, biased, n-sides.
4. Old flange, in the ESR, grounded, biased, p-sides.
5. New flange, in the ESR, biased, n-sides.
Attachment 3 v 5 is an apples to apples comparison. This confirmed that even for the best case scenario, the noise in the old flange is an order of magnitude worse. Note that the n-
sides have the worse profile, and for the old flange, the high frequency noise is almost the same size as the pulser. For the new flange, that same noise can be seen, but much smaller 
magnitude.
Grounding on the new flange made no appreciable difference. But in general, we think connecting the electronic ground to the ESR is the best decision.