Transformer Secondary Discontinuity
I do not suspect transformer core saturation. At a fixed input voltage to a transformer, you can short circuit the secondary, and the transformer core will not saturate. So, if your core does not saturate with open circuit secondary, it will not saturate with a load. (Assuming fixed frequency).

I was thinking more along the lines of the inductor load Not saturating, rather, the load seems to go to a higher impedance state at zero crossing. Perhaps indicative of a gap closing on each cycle (resulting in a higher impedance). I could imagine the core vibrating at frequency, alternating a very small gap and closed gap. Note the secondary volts increasing , indicating higher impedance load.

Edit 1: Regardless of my mentioning inductor gap, I suspect the anomaly is in the load.