That is a gorgeous guitar :) Love the colour, the flame, and the shape. I wish I could be of some specific help about which BKPs you should choose, but I really have no experience with them (yet!).
I can tell you, though, that my friend has a Framus Dragon, and plays Nailbombs through it, and has never complained about a lack of harmonic content, so I think you're right in suspecting the pickups are the problem instead of the amp :)