@prep:
I would not hesitate to recommend the Stratus for the HD800. It's the best out-of-the-box solution.
The BA with the KR PX4 (most common output tube delivered) with the HD800 may be a bit rough. Of course this depends a lot on personal preferences. I know people running 300Bs (KR, EML, etc.) instead of the PX4 with the HD800. When I had the BA, I went on a special journey to the end of the earth to acquire unobtanium Osram PX25s and performed minor modifications to the BA to run them - just for the HD800. The WA5 will also need some tube rolling to get the best results with the HD800. It's not a particularly powerful sounding amp out-of-the-box, and rectifier upgrades (Sophia Princess or similar) will be necessary to give the amp some balls with the HD800.
The BA's soundstage is much more expansive than the Stratus which is more intimate sounding (NOT to be mistaken for closed-in). Both are equally explosive on a macro-dynamic level. However the Stratus falls behind in micro-dynamics (levels of gradations and authoritative quick changes in volume, i.e. in some respects what Craig refers to as "fast") and micro-detail retrieval. Nothing, absolutely nothing I've heard comes close to the BA in these aspects except maybe the Apex Pinnacle or Manley NeoClassic (in certain configs), and even then they may not be equal to the BA. Nothing I've heard touches the BA in terms of openness or expansiveness except the T2DIY, which is a 'stat amp.
In terms of "lushness", the Woo is the most lush. Haunting would be perhaps another word for the WA5. As for the Stratus, I would not categorize Donald's amp as lush. It's pretty darn honest. But as people have mentioned, it does have a wet or damp sound. Words are always difficult. Maybe another way would be to say the that while Stratus can certainly bite hard when the recording calls for it, the Stratus always maintains an inner warmth (not to be confused with syrupy and slow.)
Now the Stratus can be made to sound more lush with the Chinese 2A3 tubes, but it never quite approaches the WA5 in lushness (when using Chinese 300B tubes of a similar style.) The side effect of the Chinese tubes is that bass becomes soft and the slightest bit mushy (but this is a very minor criticism). The EML 2A3s do tighten things up, but they drier. The BA's bass, while ever so slightly rounded, still tighter with better definition (with KR PX4) than the Stratus with the EMLs. Given equivalent types of tubes, generally the Stratus is between the BA and WA5 in terms of lushness, but closer to the BA. The BA is not lush at all.
The Stratus has the most inner warmth, followed by the WA5, and then the BA, which just has a hint of it. Because of this, the Stratus is the most forgiving in terms of matching appropriate headphone. Is there a downside? It's possible you may tire of the Stratus' "intimate" sound (I believe that's the reason one person was selling his).
All three are good amps and I could easily live with any one of them. If I were forced into one word descriptions:
Stratus: Intimate
BA: Grand
WA5: Lush
None of these tubes amps are colored to the point* where I would want to commit suicide or throw the amps out the window. And trust me, there are plenty of such amps around.
*That is poster childs of bad tube amps which perpetuate the myth, especially to noobs and minions, that tubes sound syrupy and slow; and that moderate levels of even order harmonic distortion sounds good. It certainly does not. Dig up some good recordings of Frank Sinatra or Nat King Cole. They all sond fast, tight, explosive, detailed, real. Yeah, they didn't use transistors back then when recording.
Can be bumped to 13V on higher rails but the slew stays slow 
