It does depend on the type of powder, and the amount of it. It also depends on the payload being moved and the characteristics of the payload, ie crush section of the wad and weight of shot.
"The area under the curve" will determine final velocity, the slower powders peak later (thus they are called "slower") but not anywhere as much as you have described.
How far 'down the barrel' is down the barrel? The shot charge starts someplace between 2" and 3.5" down the barrel already.
Curves and traces from Winchester are presented in Butler "The American Shotgun". Corrected 'travel distance' of all loads in that book show less than one inch when the peak is over.
The average seems to be that the peak happens at about .5 millisecond, the shot charge has moved just less than an inch, and at that point it's going about 400fps.
You tell me how far 'down the barrel' that is. For a 3" shell it approximates 4" I suppose from the breech.
The salient lesson from all this is that the peak pressure happens in the chamber, thus "peak chamber pressure".
