I own a full size maxed out KGSSHV and only heard the BHSE in meet conditions, but with the SR-009 I actually prefer the KGSSHV. The touch of warm of the KGSSHV is good for the pairing. I didn't know what tubes were in the BHSE. The SR-007 is no doubt better from the BHSE.
Hope I can shed some light. I spent some time at David's listening to his K-01 feeding my KGSShv and his BHSE with both my previous 007s and my current 009s.
I have to say I much preferred the 009s with the BHSE. It has more of everything, less smear and more more transparent with the sound IMO. Yes the 009s are brighter, but that is only part of the story. IMO the 007s (MK2s) cover up some detail and slow things down. In some systems with brighter sounding DACs this may be preferred. If the source is really really smooth, detailed but smooth, no edge or digital fatigue, then the 009 nails it to me.
On the KGSShv, I have said here before, but I thought my KGSShv (off-board full size) was as good as the BHSE. It had a different signature, and had strengths in the areas I like. I could live with either very happily, and would not say I preferred one over the other. Oddly the KGSShv does have a warm slant, almost tube sounding but a vice like grip on the bass and dynamics. The BHSE is slightly brighter and more air around vocals and instruments, and possibly the reason it needs tube matching with the 009s. If it works out of the box with the 007s, to me it is because those phones are hiding things, not because they are better. But get the BHSE and 009s right, and it is probably the best sound you are ever going to hear IMO.
The PS-Vanes in David's BHSE were very nice. I expect anyone who wants to fine tune the sound of their BHSE with their source will try different tubes. I have done that in my DAC. It does allow a bit of tuning to balance the signature of the final sound you will get. When you get to this level it is all about system synergy IMO.
Otherwise, you end up buying and selling forever. Anyway, that is my view, no doubt others won't agree. I think how much money I have spent to get this far, it has to be a logical path. IMO there is no perfect component to just do it right, it is about the sum of those separate components that work together, and also why this hobby is so good, so many variables, or maybe why it is a nightmare? I hope not.