The bridge should have exactly the same number of wires as the neck (red, black, green/white soldered, and bare), which is a 4-conductor configuration. You don't need the black wires that are attached to the base of the pickups. The neck pickup should be flipped around, when it is correctly installed, the wires should exit the pickup from the opposite corner to the bridge pickup.
You might find this diagram helpful:
http://www.seymourduncan.com/support/wiring-diagrams/schematics.php?schematic=2h_1v_3w, but the colour coding is different to BKPs:
Seymour black = BKP red
SD green = BKP black
SD white/red =BKP green/white
The schematic I linked should be sufficient, but, for completeness' sake, I'll describe it as per your diagram:
-Connect the red wires to tag A and C on the 3-way.
-The black and bare wires from each pickup should be soldered to the top of the pot (follow the SD diagram - don't solder them all to one spot).
-Connect tag B on the 3-way to a solder glob ontop of the pot.
-Connect tag D to the left connection on the pot.
-Connect the middle tag on the pot to the hot tag on the output jack.
-Connect the top of the pot to the ground tag on the output jack.
-Earth the bridge by connecting it to the top of the pot.
