In short, no. Anton LaVey had plenty of criticisms about both the Noe-Pagan movement and the occult world, as he mentions in "The Satanic Bible". If I had to call Satanism a "branch" from anything, I'd say it has much more to do with philosophers like Nietzsche, Rand, Mencken, etc. (all openly acknowledged by the religion as influences) rather than occultists like Aleister Crowely, Gerald Gardner, or Sybil Leek.
Granted, there is the creative use of past pagan mythology (The Satanic Bible lists "The Infernal Names", examples of deities from past mythologies with characteristics related to the philosophy), and a similar use of traditional "occult" practices (rituals with props, the notions of throwing a hex, etc.), but all of this is viewed in symbolic way that's quite different from the other contemporaries. Satanism is non-theistic.
Having said all that, it's not unusual to find LaVey's books in the store's "Occult" section. I guess it's just a convenient category for most people.
In ancient Jewish tradition Satan is simply an angel doing the work that God assigned to Satan to do.
The word Satan means challenger. With the idea of Satan challenging us, or tempting if you will. This description sees Satan as the angel who is the embodiment of man's challenges. This idea of Satan works closely with God as an integral part of Gods plan for us. His job is to make choosing good over evil enough of a challenge so that it becomes clear to us that there can be only one meaningful or logical choice.
Contrast this to Christianity, which sees Satan as God's opponent. In Jewish thought, the idea that there exists anything capable of setting itself up as God's opponent would be considered polytheistic or setting up the devil to be an equally powerful polarity to god or a demigod.
Oddly, proof for The Christian Satan/devil mythology is supposedly found in the ancient Jewish texts that were borrowed to create the bible. One can?t help but wonder how Christians came up with such a fantastically different interpretation of Gods assistant Satan in their theology.
Other hints about Satan?s role in human relations can be seen if you look at the name Lucifer. It?s meaning in the original tongue translates as Light bearer or light bringer. Essentially the bringer of enlightenment. The temptations of the Satan idea bring all of us eventually into Gods light. Hardly the Evil entity of Christian mythology.
I know it is not a branch of paganism, but beyond that, I really don't know.
Edit Add: In the book of Job, "the Satan" is like the "loyal opposition" in politics. They serve a purpose to challenge man to become better than he his.
That is not true throughout the Old Testament, however. Keep in mind that "the Satan" does not occupy (or need to occupy) much of the material. The people tended to be disobedient without a whole lot of help from others.
2ND Edit Add: Paganism worships (reveres) nature and sees it as neutral, being neither good or bad. People need to understand what they readily classify. That's the main reason why I won't declare "satanism" (the worship of Satan) to be a branch of the occult, though I think it is.
And I'm not interested in learning more about the occult. I am interested in learning about the religion of my ancestors (Celts and Picts), but few know much about it (because they left no written records).
Depends on the type of Satanism and your definition of "branch."
Many (but not nearly all) Satanists incorporate magical thinking into their beliefs. That definitely makes them a part of occult thinking, and they generally draw on older occult ideas.
"Paganism" simply means non-Judeo-Christian. There really aren't branches of it. Something either is or it isn't. So technically at least theistic satanism would be pagan. However, there are tons of religions who don't like being associated with that term (Hindus being one example) as it is pejorative.
People who self-describe as pagan are generally Noe-pagans, which many theistic Satanists would also technically fall into. However, they rarely self-identify as "pagan".
LaVey's organization could be described as pagan and occult. (The two categories are not mutually exclusive.) It doesn't matter to me whether members of the Church of Satan do not like these labels. The fact remains that the magical practices described in the Satanic Bible implicitly acknowledge some utilization of natural forces. This is where the paganism comes in. The fact that there is magic within the Church of Satan easily places it in the occult category. Atheism is beside the point. The facts speak for themselves.