You can get an S12, but do remember the magazine of such a firearm needs to have a bullet button installed or otherwise be a fixed magazine.
AOWs are not exempt from the California "Assault Weapon" laws.
An AOW shotgun likely meets the definition of pistol under California law.
Under California law what is exactly a pistol is poorly described and includes a firearm with a barrel less than 16".
So being an AOW may make it not a pistol by federal law, but it could still be a pistol under California law, as well as an AOW.
This has two major impacts on your AOW:
The first is that it must meet an exemption to the California roster of approved handguns, because you can be sure it is not an approved handgun.
Yet likely meets the definition of handgun per California law (while under federal law it is one type of gun only.)
That is not particularly difficult if it is large, it just needs to be a certain minimum length.
A large single shot pistol with a 10 1/2" overall length and at least 6" barrel is exempt.
So it needs to be imported as a single shot, or be brought in by someone moving into California from out of state to legally arrive in California.
(b) The provisions of this chapter shall not apply to a
single-shot pistol with a barrel length of not less than six inches
and that has an overall length of at least 10 1/2 inches when the
handle, frame or receiver, and barrel are assembled.
Once you own it then you can modify it to be semi-auto, and it no longer needs to meet the exemption criteria, but to be transferred from an FFL to you it must meet that exemption.
You are also permanently subject to this:
(4) A semiautomatic pistol that has the capacity to accept a
detachable magazine and any one of the following:
(A) A threaded barrel, capable of accepting a flash suppressor,
forward handgrip, or silencer.
(B) A second handgrip.
(C) A shroud that is attached to, or partially or completely
encircles, the barrel that allows the bearer to fire the weapon
without burning his or her hand, except a slide that encloses the
barrel.
(D) The capacity to accept a detachable magazine at some location
outside of the pistol grip.
(5) A semiautomatic pistol with a fixed magazine that has the
capacity to accept more than 10 rounds.
It can never not have a bullet button installed, because it must legally have a "fixed magazine" or it becomes an illegal "assault weapon". So you would have to carry around something extra to press the button to remove a magazine.
There is no AOW exemption to assault weapons.
It can also never have a magazine larger than 10 rounds inserted or it would be an illegal assault weapon. Of course since you cannot legally have such a magazine transferred to you anyways due to other legislation that is a moot point.
Finally something like a Saiga with a magazine inserted at a right angle will never be as sleek and easily stored loaded as a tube magazine. You also probably won't be able to make a holster for it that it can smoothly be drawn from. While any tubular shotgun can easily fit in a scabbard or holster while loaded.