The LCD-2 is hard to drive compared to some headphones but it is considered an easy load as compared to the Hifiman HE-6 (which, like LCD-2, is an ortho). James did say that the BHP-1 could adequately drive the HE-6 (but did not elaborate). If so, it should have more than sufficient power to drive the LCD-2. Whether the headphone out of the BP26 can drive the LCD-2 has never been discussed so perhaps James will give his input. I am, however, of the opinion that to optimise a pair of top class headphones, one needs a top class headphone amplifier - which I hope the BHP-1 will be one.
Also, as far as I can remember (I follow this thread closely), there's no mention from James ever that the BHP-1 would run on battery.
Hi Zeeman, after further research I think external PSU (MPS-2) will be a better option for me vs. one with battery supply. Battery life expectancy is one consideration but will have to worry about recharge time, use time, etc. and if/when the head phone amp shuts down for recharge - I will be stuffed! 
James - the LCD-2 (R2) is relatively hard to drive. Did you made any comparison on with BP26 headphone out? Any significant improvements?? Here a link on the headphone details - http://www.audeze.com/driver-update-and-leather-headband