Hello Jarrett.
I use Sowter transformers as per their recommended schematic, with lower values of both resistance and capacitance in series across the output. Parallel to that is a range of possible load resistances from far above to far below what is recommended. These are inserted across the outputs of the transformer. Trial and error showed the way here - obviously, highly system dependent.
My phono stage (home-brew) has airgap variable capacitors (100 to 400 pF, but will be changing to 0 to 1000 on the next iteration) across the inputs, i.e. the outputs of the SUT’s. These serve to fine-tune the sound, in a slightly different sense than the loading resistors - it’s hard to describe, but the airgaps seem to me to provide a useful sonic complement to the loading resistors. YMMD, but it may not be appropriate even to attempt it, in your system, as less capacitance is brighter. Come to think of it, have you tried ultra-low capacitance cables?
Have you tried pushing the load resistance up to 1K, or even higher? Say 10K? Loading with a transformer is a black art, as you know, and is, dare I say it, highly system dependent?
I found my KRSP to be highly sensitive to VTA and azimuth. But, of course, you will already have explored all that.
I spent so much time on my KRSP because I had to make it work - it is a considerable luxury for me.