I'm not sure about the grounding, but I tend to believe that it would not make any difference - ungrounded transformers work fine.
But here's another thought:
I tend to doubt that the current sensing coil inside the meter is stepped down according to the ratio between it's scales and the tiny meter movement current rating.
Rather, I suspect there is some current stepping down, but only part way, and a calibrated shunt is placed across the meter movement.
My thought is, if there is a substantial meter shunt there, might that prevent my third winding, the external 6 turns, from sensing anything? In other words, is a possible meter movement shunt acting like a "shading ring?"
I'm not sure. It would take another iron core, not the meter's, to find out.
Eleanor White