I agree with Ben, the Mule is an astonishing neck pickup, I love it in my PRS SC245, but it's not that great when splitting coils. It sounds too thin and weak, I almost never use it in split mode. It's not a dark pickup at all, but neither is my guitar, so if your guitar lacks high frequencies, maybe you should check even brighter pickups, as Riff Raff or VHII
VHII is a great pickup too, brighter and a bit thinner, compared with the Mule splits better but lacks a bit of "character" for my taste
I've never tried a Riff Raff, so I don't know how good it's when splitting.
I had a quite dark Les Paul with Rebel Yell bridge and Emerald neck, great combo too, I guess the Emerald should work fine with the Black Dog too, but I didn't installed the split-coil in that guitar, so I don't know if it's a good "splitter"