Great heavy tone on that clip! Very well produced.
I have no experience with 7 string guitars, but I feel that the Painkiller would be too midheavy for that kind of tone. The clip made me think of Miracle man, or possibly Cold Sweat. Depending on the amount of mids you want (CS has less, and a brighter top too), and the natural tone of the guitar.
For a fluid, defined and singing shred tone on the neck, the Cold Sweat neck model is hard to beat, but I've heard great things about the Miracle Man neck too.
Of course, the tone on the clip model is also very heavily amp-dependent, but my gut feeling is that Miracle Man or Cold Sweat would get you there. It wouldn't hurt to e-mail BKP staff for a word on the matter.
-Zaned