No, I'm afraid not.
From a RAW side, obviously no. While you can make up a chaos marine warband, call it whatever you like and paint it however you want, and then say it's splintered off from, or aligned with, or simply fights in the same style as whatever legion or renegade keyword you like, that isn't how the Servants of the Abyss work. The mechanical implementation of that legion business is via the variable <legion> keyword that none of the BSF units have.
From a RAI side, though, still no. None of the BSF units have that variable chaos marine legion keyword because, for the most part, they aren't chaos marine units at all, not even in the 'kinda sorta' way that cultists are. "Servants of the Abyss" isn't a warband/legion/chapter subfaction type keyword, it's a faction type keyword. Ie, equivalent to "Heretic Astartes", not "Black Legion." It's bespoke, catch-all faction keyword for the hodge podge of chaos stuff that was at the Blackstone Fortress during the events of that game.
There are a couple units that are both "Heretic Astartes" and "Servants of the Abyss" - specifically the Lord and couple chaos marines, but you'll note that they're also "Black Legion", which again just goes to show that "Servants of the Abyss" is not a warband/legion keyword.
So anyway, no. You can only field the Servants of the Abyss as their own separate side detachment, and that seems to be a deliberate choice on the writers' part. Further, it would not make sense to field most of those units as part of a chaos legion (black or otherwise) or even generic chaos marine detachment, as the traitor guard units and characters are neither black legion nor chaos marines, again not even in the way that cultists are.
Hopefully some time within the next year or three we'll see a proper traitor guard / renegades & heretics / lost and the damned / servants of the abyss style faction for 40k, and then presumably some or all of the BSF chaos stuff will be properly fieldable as part of that army. There have been several seeming moves in that direction - the mutant models from the recent rogue trader box, the traitor guard models in BSF, forgeworld discontinuing their renegades & heretics line, etc. Hopefully the wait won't be too long. IIRC it wasn't too long of a wait from when genestealer cults were reintroduced in that deathwatch boxed game (with a similarly restrictive side-detachment implementation in 40k) till they got a proper 40k faction release.
In the mean time though, the Servants of the Abyss are limited to being a mini side detachment of dubious usefulness. That said, the FW renegades & heretics rules are still legal out of the FW chaos index, and you might be able to put together a workable R&H detachment out of a couple BSF sets plus some assorted guard business.
Edited by Sception, 25 April 2019 - 06:21 PM.