I think the reason that you might be seeing the Low end transferring well is that you are using reasonably low levels (-17dBu/-20dbV). These are reasonable levels for most microphones.
Usually you see the distortion come in at higher levels (300mv on the little beyer's for instance) but they are all different.
As far as the mid and high end, it is interesting.
I note that the test setup did not have the transformer terminated and this can lead to ringing, especially in cheap trafo's. I also notice the rolloff starting at 4K, this could be parasitic capacitance to ground operating with high input impedence (what is the generator impedence?). Then the response curve heads up, this could be some interwinding capacitance causing signal to transmit from primary to secondary capacitively (although with the electrostatic shield I don't know if that would happen) or it could be ringing ( I don't know what a VTVM is, but I am going to assume it is a scope (I. E. it has low capacitance and no time constants we need to worry about).
A first step might be to take a look at the schematic from the mixer you pulled these from and see how the designer terminated the transformer. If you don't have the schematic or the PCB, then try some sort of resistor load or Zobel to terminate the secondary. For a 1:15 transformer start with values from 225K to 450K across the secondary. Possibly in parallel with a 50pf to 200pf ca
Before messing with that, do make sure that the source impedance is appropriate (I.E. Looks like a microphone). 150 to 350 ohms is typical. If it is too high you can get there in various way (step down trafo, preamp, pad). If the source impedance is too high, and the secondary in unterminated, then the capacitive effects will be pretty dominant.