I may have spoke too soon! (TLDR: I think the transformer was a red herring. But I did find some other interesting details on the symptom.)
After doing more investigation, there seems to be more variables at play in my situation to be useful to others. What I’m finding is that with my audio interface, the noise only occurs if the input gain connected to the Daisy is above a certain point. If I keep each channel gain right below that point, but then compensate with the master gain of the interface, I can juice the usable signal seemingly indefinitely without running into the noise. I’m not sure how to dig further into the implications of this, but it does seem to me like another case of the noise being dependent on the characteristics of whatever is connected immediately after to the Daisy.
As far as the transformer goes, I originally just grabbed a transformer I had laying around from a friend’s record player, so no part number and I don’t know the ratio. Likely a step down ratio, which probably masked the issue in my case because I had to apply makeup gain on my interface. I later tried with a 1:1 EI19 transformer and it actually made the noise worse! So… nevermind on that angle I think. 