BHSE or Aristaeus for SR-009?

1) Everything

2) Not really

Neither. I suspect the BHSE is likely overkill for the SR-009, and the Aristaeus wasn't made to amp any Stax headphones. The Aristaeus is HeadAmp's take on the Orpheus (HEV90) which was an amp for the Senn HE90.

As others have noted, the Aristaeus was voiced with the HE90 in mind primarily. It's going to be a really stereotypically "tubey" (warm, lush, etc.) sounding amp, especially compared to the BHSE which will sound far more transparent.

Whether the Aristaeus is appropriate for the SR-009 or not is entirely up to you. It can certainly drive the SR-009 without any technical problems, but some might feel an amp like that blunts the capabilities of the SR-009. On the other hand, some may feel the SR-009 sounds too thin and/or bright on the BHSE.

My advice would be to try to audition them in person somewhere if at all possible. Otherwise, read a variety of impressions for each amp you're interested in until you get a general idea of what it sounds like paired with the SR-009. Then go with what suits your personal tastes. There are other amps out there to consider: for example a Stax SRM-717, SRM-727 (modded), Liquid Lightning, or Electra. You can also go the DIY route and build / have built a KGSSHV. Also keep in mind your source is going to play a big role. You'll want something really nice to feed into a top-tier Stax chain.

Edit: Speaking from personal experience, I prefer the Liquid Lightning with the SR-009. I'm one of those folks who find the BHSE underwhelming with the SR-009. Others love the pairing. Personally the BHSE would be my first choice for amp with the SR-007.

Here's a good article by Tyll on a few well-known amps: http://www.innerfidelity.com/content/listening-great-headphone-amplifiers-and-stax-sr-900

The Aristaeus isn't there, which isn't surprising since [again] it's associated with the HE90 more.