I would try raising C11, maybe as high as 22nF, 33nF, or 47nF, for an additional treble cut. You could also bump up C3 a bit to increase the low mids.
The Timmy doesn’t have any circuit that scoops out mids. Though I agree it sounds like it has less mids than many drive pedals.
When the signal clips, it tends to exaggerate both low and high frequencies. The clipping harmonics from the low frequencies get bunched up and muddy. The clipping harmonics from high frequencies get really high and harsh.
With the gain down, bass and treble at max seems pretty flat (transparent eq). As you increase gain, you have to cut bass and treble to sound transparent.
The Timmy formula is
Bass cut ( around 1.2kHz), clipping, treble cut (adjustable frequency)
The tubescreamer formula is
Bass cut (around 720Hz), clipping, treble cut (around 720Hz), then a treble cut or boost with the tone knob.
So to make it more similar to the tube screamer, you could introduce another high cut (C11 increase) or move the low cut frequency closer to the TS (C3 increase)