I simply compared headphones like the HD650, HD 800 and HD 800 S with the Chord Dave against the Chord Dave + Taurus MKII. I don't hear anything which suggest the Chord Dave is "underdriving" these headphones. So anything which sounds different from how the Chord Dave originally sounds is colour according to my ears. Also there is no such thing as wire with gain since even the interconnects alter the signal (the sound coming from the Taurus MKII is impacted by the quality of the cables. I also heard the same on speaker setups).
I'll see if I can borrow someone his LCD-4 to test if the Chord Dave fails to drive these headphones properly.
As mentioned, please try some DSD recordings with your HD800S, fully balanced, I hear clear improvements in clarity and control (w/ my GS-X Mk2) with the added headroom. Not huge, but still there, so much so that the TT --> GS-X Mk2 is my preferred setup for my HD800S. That said, straight out of my Hugo TT, the HD800S still sounds stupendously great! I did listen to the LCD-4s from the DAVE alone at RMAF last weekend and the results were pretty clear: needed more power. While not the HE-6s, the LCD-4s need quite a bit more power than what was on tap. The folks at Chord said I took it beyond "the point of clipping" just to get it to sound sufficiently loud.