It does sound like B, W, G and bare wires are are correctly wired for a Dimarzio. It is really a Dimarzio right?
The red wire placement is probably the problem, but considering how the PAF is wired I don't see how it could be split into single coil mode or cause any hum in this configuration (B and W tied together and insulated only allows dual coil humbucking mode).
The red wire should be wired to the switch. The question is which position on the switch. It should be and end position on the opposite side of where the Carvin pickup is connected. This position would likely not have other wires connected to it.
Here is the unsophisticated way of finding the right connector. Unsolder the red wire and touch it to all of the switch connectors while you have a string ringing out. Make sure you have the switch in the PAF position. This sounds crude but it may help.
Of course there may be something wrong with the other wires. Check the wiring again. Make sure B and W are tied together and are truly insulated. Check the ground connection. If you have a multimeter you should make sure that the volume pot case is tied to the outer ring of the guitars output jack. Find out
There is a chance that you may actually be causing the Carvin pickup to generate noise or weak output in conjunction with the PAF. To find out if the Carvin pickup is active when you switch to the PAF, try checking which coils are active on both pickups. You can do this by plugging in and tapping on the each of the coils polepieces with a steel screwdriver. If the coil is active the tap will sound bright and loud. If it is off you may still here something, but it will be softer dull thud.
Let me know what happens.