Progress! Aluminum foil wrapped around the capsule body (but not covering the face of it), which covers the wire connections to the capsule from the wire, makes the hum disappear. I'll solder in some copper foil to the situation.
Is this standard best practice for PiP mics that I should have known about?
Get a mono plug. It'safer.
I think we've shown that it makes the mic go dead, though? When I connect ring to ground (which I think is what a mono plug will do, right?) the mic goes totally dead.
How is the capsule shielded? Cover the capsule with an empty can. If the hum is gone - you have improperly or insufficiently shielded capsule.
Per above, this indeed seems to be the issue!
-c