I really wouldn't be surprised to see something like this as the starting point:
Notice that it's been adapted for a single tone control. Most other sections are 1:1 with the actual Butler design.
IC1A is just an input buffer, and I could imagine Hermida dumping it altogether. I know Sean certainly would - not sure if he had any input on this. The marketing to align the thing with the Tube Driver tells me the latter - I don't think Alf would be hip to that IMO.
IC1B could then remain as it is - signal thru the inverting input, or it could be changed to be like a Rat/TS/etc. - signal thru the non-inverting input, and all the typical circuit constructs like you would otherwise see. There just would be no 12VDC regulated in and run on the traditional dual rail - it would be the 9V/4.5V/0V single rail setup found in pedals.
...At 9VDC, IC1B would definitely clip as the gain is increased. So clipping diodes would either probably be used in its NFB loop, or shunted to ground after it.
If dropping the IC1A buffer, it becomes a 2nd gain stage instead. A number of things could be done with it, so I won't speculate.
...But there would have to be some way to approximate the starved tube stages. I would think that could be handled with some sort of array of clipping diodes, but I guess I wouldn't rule out that the solid state "quasi-equivalent" - something like a pair of transistors with common emitter/common source - could be used.
The more important thing is to have the operation curve of that portion behave in "an un-tube-like manner," as is found in the Tube Driver. Looking at the following transfer curve plot for a 12AX7, it's easy to see what you have going on with such low plate and grid voltage:
...It doesn't even really show anything on that plot because you don't move beyond >+12VDC at the plates or <-12VDC at the grids. And at a swing of either 4.5VDC either up or down in the Dover Drive, the effect will be further caricaturized.
While the clips may be kind of crummy/limited, IMO they really do sound like there is no recovery stage after what would be used as a surrogate for the tube in the Dover Drive. So I would expect the "bulk of the clipping" to occur there. And some of the octave'ish stuff I hear tells my ears that you've got some "rectification effects" with diodes or whatever - that's how you tend to get those "octave multiples" with fuzzes and such.
...And it's kind of a grainy characteristic, which IMO again sort of indicates no recovery stage, so it would dump right into the tone circuit. That's why I have a link to the Baja Tube Drive instead of the Butler - just to show the singular tone control - keeps the same vibe but w/o the complexity of the passive T/M/B (or high/low) like in the Butler/Chandler/Dean Markley/etc. stuff.
So - if something other than passive diodes are used - if a pair of transistors are used for example - the trimmer could indeed be something for manipulating either gain or supply voltage. Or it could be something for continuously blending in/out different clipping diodes.
...At 10mA current consumption, the op amps/transistors chosen do not require much power to operate. That is the biggest deviation from the Tube Driver of all.
Heck - maybe Sean had much more of a hand in "designing this" than we might realize. Maybe Hermida provided the front end, and then Sean tacked on one of his many tail-end diode configurations - the things that barely separate the COT from the Les Lius, for example. Or maybe Alf helped him with those. Maybe it's nothing more than op amp stage(s) => BJT stage => clipping diodes => tone circuit. Or something like a Micro Amp into an Electra Distortion with a tone control.
...The "big differentiating design feature" with most of the Electra Distortion-based variants that Lovepedal/Gaspedals/Greer/Earthquaker Devices/Emerson/etc. seem to come up with is a switch for different combos of clipping diodes. So you would just replace that with a blend pot - at one end, you could have a pair of LED's, and at the other end you could have a pair of germaniums/Schottkys/etc. on that "bias" trimmer.
I think it's fun just to speculate.