I finally got around to swapping the P pickup out of my PJ
It wasn't that I didn't like it, I did, it's more that it wasn't quite the tone I'm useing these days.
First I put a Fender 62Ri Original in. This was just laying around and I didn't want to work on two basses so was going to leave the Fralin in my CV.
A very strange thing happened with the fender. First off it sounded great. Lots of volume and equal to the bridge pup. When I had both on the sound was very low and holler sounding. Now I know this meant one was out of phase, so I swapped the wires over on the P, and the tone and volume was much better with both pups on, but now the p pup was softer and with less punch on its own. I've not encountered this happening before.
I decided to give the Fralin a go after all and it went well. It's a bit fuller sounding, and seems a better match to the bridge pup.
I put the fender pup in my CV and it's working fine. I even swapped the wire over to see if that made a difference and it didn't. Very strange.
So, that's one pup down, one to go. I've decided to go for a Fender Super 55 jazz pup. I've not read much about these but what I have read is positive. I really like the stock jazz pup, but it's buzzing quite a bit, and we tend to play some pretty crappy dives where the electrics aren't the cleanest, and so I expect a lot of buzzing at some of them.
I learned through many botched attempts with a VM Jag (I had many of the same issues you describe) to leave the stock pickups in Squier p/j config basses. They just don't seem to play well with swaps. I find better luck with my sound through pedals and amps. If I'm recording, I just bring my '75 P.