I think it's worth flexing the PCB that the coils are mounted on, to make sure that there's no signs of the bobbins being loose. I had a very similar issue and it turned out that the small amount of uneven flexing that took place whilst I was setting my pickup heights had caused two of the connection to become loose, giving me very similar (and intermittent) symptoms. I appreciate that you've checked for continuity already, but it could be that you have good contacts while the pup is on the bench, but the connection(s) might be cutting out when the pup is re-mounted and under spring tension in the guitar.
In my case, I ended up squeezing the coils onto the PCB whilst I dabbed each of the connections with a soldering iron. With the contacts re-made, I never had a problem again.
Just be careful whenever you touch the pickup height / mounting screws - work in pairs (both treble sides or both bass sides) to avoid any twisting of the PCB.