Roland / BOSS have a thing called Tube Logic. I'd forgotten about it. It's some clever stuff that makes power transistors (or more to the point, an Op-Amp) sound more like power valves. More. Like. Not "exactly like". This is rock 'n roll. Nobody can hear your nuances over the drums.
In a last desperate attempt, to do something about the ineffably "blurry" clean tone I was getting from the humbuckers on the 594 with the power selector at 25W, the Master volume at 09:00 and the pickups at 6 / 7... I turned the volume to 12:00 and dialled the guitar volume back to 2 to bring the volume back to polite levels. It sounded almost identical to the first setting, but, I don't know, 10 lbs lighter?
Then out of curiosity, I turned the power selector to 0.25W (which takes all of 6dB off the 1W loudness of the speaker), and dialled the guitar volume back to 8. Oh Holy Moly! When played through an HX Effects channel with a Tube Screamer, or another distortion pedal, the clouds parted, and the sun shone through. Okay, we're not talking Mediterranean, but it was good enough.
You'll notice this is entirely counter to how Real Amps work. To get a clean tone on a Real Amp, keep the volume / gain below a certain level, and crank it up to get distortion. Cranking up the guitar, while turning down the amp, to get distortion, or cranking up the amp, while turning down the guitar, to get clean, is just being silly.
But that's Tube Logic for you. (Plug in via the Power Amp In socket, to by-pass the K's DSP, which you don't need because you've got a multi-effects pedal.)
It works for me. (For now.)
I do need to dedicate three blocks (two EQ and one compression) of the nine in an HX Effects circuit to what amounts to managing the Katana so it sounds vaguely like a proper amp, but I can live with that for the moment.
There's one combination I'd like to test: a Big Trees
through a 1x12 cab. Not cheap, but then neither is a decent home hi-fi. And I think it looks cool.