To begin with, the HD800s have VERY high impedance, which means less than 20% of the headphone amps in the world can drive them to anywhere near their potential.
The Bottlhead amp is not one of the amps I would recommend for HD800s without modifying the output stage for high impedance headphones (requires serious re-engineering).
On the other hand, the physics of the ZOTL output stage make the mZ2 ideal for high impedance headphones like the HD800s.
At our Head-Fi meetup recently one person had a Bottlehead amp that we compared to the mZ2 on headphones that are much easier to drive, such as Ether Cs or the upgraded Oppos.
The Bootlehead amp sounded warm, beautiful, and intoxicating, BUT far less resolving, far less extended, far less tight, and far less natural than the mZ2.
If you want a romantic sound that covers up all the flaws in a mediocre source and bad recordings, go with the Bottlehead amp.
If you want accurate time, tune, tone, and timbre, along with extension and dynamics, then I would recommend the mZ2.
Of course with an mZ2 and HD800s (my last year's love) you will hear EVERY flaw in the recording and your source. You're talking about a "Ferrari" system that "feels every pebble in the road."
In general, with HD800s I would recommend upgrading the cable to that Peacock MkII cable or some other cable with a warmer tonal balance, as well as upgrading the stock tubes in your mZ2 to NOS tubes.
On the other hand, the Bottlehead amp with headphones that work well with it (not HD800s) will make almost any source and any recording sound lovely.