These are two of the best choices available for what you want to do, but you should also add the TTVJ 307A amp to the mix. The key is that all three are great across the spectrum of headphones you might ever want to use: low impedance, hi impedance, and K1000. From my personal preferences, the 307A is at the top, with the BA being a very close second, and the WA5 third but still a great amp.
I think the WA5 with Western Electric 300Bs is a great sounding amp, and I prefer (from audio memory at least) the sound of that setup over the sound from those new Shungang Treasure tubes that were in the WA5 at CanJam. The level of detail, clarity and presentation are all great for a single-ended tube amplifier. It does not drive the K1000 as well as my Moth 300B with the same tubes, but that is a speaker amp only which is best for them, imo. On balance, the way I (and others) have seen the WA5 is that it is a Swiss Army Knife of amps and does everything very well, but nothing the absolute best available.
The 307A and BA both add in balanced drive and suitability for pretty much all cans. The 307A is probably slightly drier and more transparent, while the BA has a slightly warmer and full-bodied sound. Both are very transparent to the source, and neither of them crosses the line into too cold or too warm to my ears, but there is a definite difference in sound signature. If I were pressed, I would say the 307A is more neutral and can give you an even truer representation of the source. The 307A adds a gain selector that goes right down to IEM levels and I think that it actually succeeds even at that level. Both of them drive the K1000s well and easily, both of them sound great with the HD800s from short listening at CanJam (307A) and our recent meet (BA).
The last time I heard these two side-by-side was when the BA was in a prior prototype iteration, and I have been wanting to recreate that comparison sometime. The BA came in second then, but hearing it at CanJam and Burlingame its final production version is way better and it would be a harder decision.