Do some digging on this thread. Also the Carbon and 009 thread and at the 'other' place.
Also ask Purk and Mulveling who have / had both. My opinion is they are close but different. The BHSE is a chameleon to an extent with tube choice. As stock it can be a bit bright on the 009s, but fine on the 007s.
The Carbon 'may' have faster bass and soundstage width. It is difficult for me to say exactly as I am relying on my memory of the BHSE and 009s in that demo about 11 months back. Also there are a few slightly different builds of the Carbon, some with the 2 x 15V LV supply boards, others that take that off the main PS board. And possibly the volume pot type may add to a (slightly) different sound. The PS on the Carbon is a later design than the BHSE so 'may' be better. You will need to ask JimL or Kevin G for details on that.
If you get a decent builder he can advice you.
Don't forget, we are talking summit-fi in headphone land here, and you need a clean source (good smooth sounding DAC) or fatigue may entail. IMO it is not a good idea sticking the 009s and Carbon or BHSE on the end of a basic Laptop and cheapish DAC, it won't end well. You get so much speed and transparency with this system you might not like what you here (in your DAC and front end). The same applies to 20K+ speakers, with transparency there is a price, but get it right and it is heaven.
Good luck.