So did a quick A/B of the Shaw Tube Injection and the Broughton P-15 to compare the extent to which I could get the tube to break up.
I played at low/moderate volume using a passive bass into a pedal EQ into a VCA compressor into the preamp to a class D power amp.
For the Shaw, I had the pad wide open and the volume on full, bass and treble at noon to 1:00. I did not hit it with crazy loud volume, but it did get a strong clean signal. I could not get the Shaw to break up, as I understand that term. That doesn't mean it didn't sound great, or that I couldn't use the fat switch and different mid settings to create more gain and tonal excitement, harmonic content, whatever you want to call it. Just not tube break up that, when things go right, you can try to control with your playing style. One thing to note is that the Shaw is not Baxandall-based, so it has less of an Ampeg tonal vibe to it to my ears (not worse, just different).
In comparison, I could easily get the P-15 to break up by turning the volume to about 3:00 and beyond. I won't go into the details of how it breaks up because, understandably, that will just waste your time and patience.

But if it makes you feel any better, to my ears it too ultimately lacks what the interaction of power tubes brings to the table.
I'd talk to Shaw about your goals and see if his regular Tube Injection, or perhaps an easily modified one, would fit the bill for you. He and Josh seem to be the builders of what you're looking for. You could also try talking to the folks at 65Amps (in LA) and see what they might come up with. Their 60 watt Apollo is B-15 inspired and very nice (and also discontinued). Give a shout if you're in the Bay Area and want to check something out.