I have a BHSE aswell as the SR-009's, and the other amps I've heard with my 009's are, LL , SRM-007t and the SRM717. At the time I heard the LL it was along with my BHSE and to what I can recall ( when I heard them both together was about 18months ago ) the SQ from the two was very similar. Build quality wise though, IMO the BHSE is in a completely different league.
Both the BHSE and LL sounded better than the two Stax amps driving my 009's, but out of the two Stax amps I thought the SRM-717 sounded better than the SRM-007t, however, a friend of mine which owns the LL and owned the 007t + 717, has also heard the Stax SRM -727 with his 009's, and thought his 009's sounded best with the 727 out of the Stax amps.
As you know there's a possible long wait time for the BHSE, so if you don't want to do that and you're only looking for SQ and aren't bothered about build quality, then IMO the LL would be a good choice, if I was in your position and I didn't want to gamble at the wait for the BHSE, I'd go for the SRM-727.
I would say that the differences between the KGSSHV and LLMK1 was quite vast (i.e. the LL didn't fair very well):