With BBF pressed, BBF overrides FPS when the shutter button is pressed even if the shutter is set to AF+AE lock.
Yes. I also noticed that the override holds as long as you keep the Fn (BBF as you call it) pressed, even if you take photos in between. So you can focus once, and then keep the focus lock over multiple pictures as long as you keep the button pressed.
My definition of BBF in the above was BBF only. If you read articles about BBF (or BBAF), then the main point is disengaging AF from shutter so the camera doesn't decide when to focus, you do. But yes, the terms are confusing.
With BBF not pressed, FPS then works when the shutter is set to AF+AE lock.
That's my experience. So you cannot have FPS if your shutter button is set to AE lock only.
It's the same with manual focus. If your camera is in manual focus, or focus at infinity, or any other non-AF focus mode, FPS doesn't work. So my conclusion is that for FPS to work, you have to have the camera set in the way that half-press of shutter button activates AF.