I've had both sets of pickups, so can give a little insight.
I found the CS bridge to be very tight and clinical. I play vintage Marshall style amps with gain levels from clean to 80s hair rock, and I found the CS too sterile. For me it was really suited to the heavier end of things but was too cold for clean and crunch tones. As you would expect really, having a ceramic magnet.
The CS neck has an Alnico V and from what I can remember, it's similar to the HD neck. Quite open and airy, and not muddy.
I would guess that you'll find the biggest difference is in the bridge pickups, not the necks. I know exactly what you mean about the openness of the neck HD in comparison to the bridge, and this is one of the reasons I like it so much! It doesn't get too thick sounding and is a nice change in tonality from the bridge.
If you're happy with the HD bridge, something you may want to consider is sending your neck pickup back to BKP for a rewind. It sounds like you may want a few more turns of wire on there to fatten it up a bit?
I'd give the guys a call.......