After quickly reading the thread, my initial thought is that somehow the signal source you're using is "floating" (perhaps a generator that's not grounded via its AC power) and has considerable voltage with respect to this gear - which would appear as common-mode on the transformer primary. Of course, with the primary CT grounded, the CM voltage would see very low impedance (same current in opposite directions in primary) but, when the CT is ungrounded, would couple to the secondary in spite of the pri to sec Faraday shield. I'd start by looking at your signal source with a scope grounded to the unit under test.
Incidentally, the difference in DCR of the secondary "halves" is likely due to this being a 3-layer design. The first layer is half the secondary (the lower DCR part), then a Faraday shield, then the primaries (possibly bifilar wound), then a Faraday shield, then the other half of the secondary. Since the halves need to have the same number of turns, but the winding diameter is bigger for the outer layer, the DCRs will be quite different.
Incidentally, the difference in DCR of the secondary "halves" is likely due to this being a 3-layer design. The first layer is half the secondary (the lower DCR part), then a Faraday shield, then the primaries (possibly bifilar wound), then a Faraday shield, then the other half of the secondary. Since the halves need to have the same number of turns, but the winding diameter is bigger for the outer layer, the DCRs will be quite different.