The BBB system CAN incorporate your chlorinator. It's more of a philosophy of pool care, and is VERY flexible. I don't have an inline chlorinator, but I do, occasionally use 3" tri-chlor pucks in a floating chlorinator, when I need a steady stream of chlorine, my pH is higher than I like, and I need to add stabilizer/CYA. But I don't NEED to use them.
I can do EVERYTHING I need with Bleach, Borax, Baking Soda, Muriatic Acid and straight CYA granules.

Chlorine is chlorine and you can use whatever makes your life easier that day. But you can always do it all with the 3 B's of BBB + Muriatic Acid and CYA powder.

But what the BBB system teaches is you do NOT need most algaecides, Nature2 or other "no chlorine" systems, flocculent, anti-foamers, clarifiers. or Baker's Yeast (I just threw that in).

I have used and will use again at some time, pretty much every kind of chlorine, but that's mostly for MY convenience. I currently use an SWCG but still add bleach as needed--I have a "salt" pool.

Again, BBB is a way of thinking, of approaching pool care sensibly, logically, easily, without the mumbo-jumbo. And it works.