I actually don't have hardware in hand; I'm deciding between buying several controllers, some of which lack sliders. So I don't know exactly what the button would send!
As far as what Kontakt wants, I'm finding that difficult to ascertain as well! Previously I've been sending CC values from a physical slider via "Map Parameter," but I don't know if Kontakt is responding to the CC directly or if something's going on under the hood in Mainstage.
I think you've already helped a lot just by pointing me in the direction of Scripter.