I think you have to remember these transformers were designed NOT to color the sound. So while they contribute some non-linearity that is easily measurable they shouldn’t change the audible results much if at all.
Now you could design a system that purposely drives the snot out of them. Or you could just get a fuzz box
The less sane you are about driving them, the easier it is to saturate the core
But I think an important point is that the distortion is inversely proportional to frequency, which is why my preferred transformer abuse is with bass. Also, the distortion caused by hysteresis, which doesn't depend on excessive levels, is also in the low frequencies. I don't hear much difference in mids on up. For example, the average transformer doesn't do much with voice, as far as I can hear; if someone wants to use a transformer to change the vocal quality, they'll probably be disappointed.
Curiosity - what transformer are you using and is it just the transformer or is there a circuit with other components?
A couple posts back I mention that it seemed like there was some compression, this is because running straight in with the bass and the same setting on the BTC - bass cranked and treble attenuated slightly - and it was super bassy (as we would expect). Using the transformer(s) it sounded more "even" with the bass attenuated a bit and the trebles audible but not to the point of revealing my less than perfect bass technique.
I looked inside the Whirlwind IMP 2 and there are two 1/4" TS jacks, 1 XLR male output jack, a transformer and a ground switch. The Radio Shack DI has a schematic drawing on the label, just the jacks and a transformer. I am assuming those bore the brunt of the bass and BTC since they were next in the signal chain.
Now I need to dig around and see what other cables I have, I remember seeing an XLR to 1/4" TS cable that I need to try to get the bass and BTC to go directly to the Beyer. Otherwise I can't really tell much as to what it sounds like. I may have to change the XLR plug, depending on if male or female. I have the parts.
If that is feeble I'll suspect that the Beyer really wants to see a microphone or similar load at the primary instead of whatever is coming out of the BTC (or the bass, which I will also try).
While I am by no means sane, this isn't exactly like running with scissors - "You could put out an eye!!!!!."