Hi Alex! Unfortunately, I think you are limited by the maximum DC current the interstage will accept. As Ralph mentions above, SET transformers, whether interstage or output, have air-gaps in the core so they can tolerate the DC offsets. This has the side effect of requiring much larger cores, since air-gapping the core reduces inductance several times. OK, so the IT grows in size. Unfortunately, there is another downside as the core gets larger ... stray capacitance between windings also gets larger, which is a fancy way of saying less bandwidth.
So the effective cost, assuming clean-sheet transformer design, is decreased bandwidth on both ends of the spectrum. But transformer designers are clever and now have access to computer modeling that wasn’t available in previous decades, so clever interleaving schemes, and analysis of flux density in different parts of the core, lets them "do the impossible" compared to 20 or 30 years ago. So I would suggest looking at Cinemag or Monolith if you want an exotic custom design, but I warn you that requiring 60 mA as well as wide bandwidth will push the transformer to the limits of what can be done. And it will not be cheap ... custom work never is.
Don and I are using both Cinemag and Monolith, and they are custom designed for us. They are not off-the-shelf parts, and we collaborated with the transformer designers over several iterations before we got to where we are now.
You are limited in scope as long as you are buying off-the-shelf transformers, whether IT or output types.
However, it isn't bad as it sounds. Even a massive tube like the KT88 works just fine at 36 mA. Conveniently enough, all of these family of beam-tetrode tubes have the same octal socket and the same pin wiring. So just twiddle the cathode resistor to run the tube at 36 mA, and off you go. And if it's a little guy like a 6V6, just run it around 25~30 mA, and compare your favorite types. Frankly, the choice of cathode bypass cap may be more audible than the tube you choose.