And pins 2 and 3 of the XLR connect to the black and green wires from the microphone output transformer? Or black and blue?
The SK-50 should have two windings so yo can switch between reasonably low (~200 Ohms) output (black and green) and 15k imedance (black and blue, which is too high to drive a cable of significant length and also noted should be connected directly to a tube grid, not through a transformer).
The manual does point out that grounding terminal 1 is only for unbalanced connection, not balanced.
The cable shield connects to pin 1 at the microphone end (which you already noted is connected to the microphone body) and no where else inside the microphone? Have you verified that the connection to microphone case has not become corroded over time and become high resistance?
And the cable shield connects to terminal 10 at the pre-amp?
The manual for the RCA microphone does specifically note that the preamp input impedance "should exceed 2000 ohms to avoid attenuation of the low frequencies."