I used to own a BP26/MPS2 and used it with various headphones (Grado PS2, HiFiMan 500). Certainly no slouch when I ran my headphones through the BP26 directly and honestly was better than my dedicated headphone amps I had at the time time (a Grado, HiFiMan EF2). I sold the pre-amp last year when I downsized my system, (also sold my 4BSST2) and bought a BHA-1 just for headphone listening with my set of Senn 800s.
I never compared them side-by-side (and I spent way more time listening to my BP26 through my speakers than my headphones), so take the my observations with a grain of salt. But the BHA-1 seems to handle the job with greater heft and control. I don't know how the BP26 headphone stage circuit is designed, but the BHA-1 is pure class A as I recall and it does seem to me to have a sound that I would equate to the best class A amps, i.e., full sounding without being diffuse, able to bring out great micro-details if the source has them, no background noise, never challenged by big dynamic swings in the source.
The one thing that I do notice is that the volume control works very differently and I have read this is by design. There are threads that explain the differences between how the BP26 (and most pre-amps) are wired with respect to volume control vs the BHA-1. The differences are real and I had to get used to setting the gain switch on the BHA-1 to accommodate different input configurations.
Lastly, I'm now going to put the shoe on the other foot, i..e, test out using the BHA-1 as a pre-amp. I just purchased a class D amp and a set of small studio monitors for my small room and plan on driving the amp via the BHA-1 XLR outputs.
Should be interesting....