OK. I think I've worked out what it might be, and it's a design issue with BKP P-90s that affects all who attempt to use it in a 'batwing' (full pickguard) SG with this type of mount.
The BKP P-90s have a pair of what looks like rivets on the back of their baseplates where two screws fit into the factory pickup that go through a block and hold the 'batwing' mounting plate in place. I had forgotten about this but when I first installed the pickup I had to leave the block and those two screws out, so that the pickup is mounted only by a pair of threaded bolts and nuts through the holes that normally use the long screws that in most soapbar P-90s are then screwed into the wood of the guitar.
To fix this it would involve sending the pickup back to BKP and at the very least removing the magnet and ensuring that it has the correct screw threads in place, and then sending it back to me.
I haven't heard back from Ben yet.
You can see what I mean if you look at the photos below.
The BKP (sorry the black rivets didn't turn out that well in the flash photos):


My stock pickup and the mounting system:






EDIT:
I think I might have the guitar modified so that the pickup screws straight into the wood. It will leave me with two adjustment holes in the pickguard but I plan to replace that discoloured pickguard anyway. That form of attachment will give a much more solid connection. Here's Jason Lollar installing a P-90 in an SG, and he discusses the batwing plate issue.
https://www.youtube.com/watch?v=AjOKO6D5puA&list=RDAjOKO6D5puA#t=1