looking for a good P/J

[Kwesi]

Spector Euro.

Beat me by a hair, KPax.

+10000 they give a more modern sound thanks to the EMGs and Tone Pump pre but I guarantee you it will handle damn any genre with ease and tone to spare. If you can save up maybe $300~500 more then find yourself a Lull or Sadowsky I've tried both and they absolutely KILL! I was digging on the Lull a bit more because the preamp seemed to be able to shape the sound to a greater degree than the Sadowsky.

 

[-Kiedis-]

Yamaha BB614.

 

[winston]

I must agree that Spector nails the classic modern P-J tone to my ears.
The G&L SB-2 is also great for more aggressive types of music.
You could probably find a used Pedulla P-J Rapture or MVP in that range for warmer, rounder sound.
Fender Hot Rod Precisions from the early 2000's are great but quite a bit mellower than a Spector.

 

[Nedmundo]

For well under your price target, you could get a Fender Reggie Hamilton Standard Jazz, which has U.S. pickups including a noiseless J. It has active/passive operation too, which is rare with Fenders. These are really great basses, but some can be heavy.

If I frequently used the J pickup on a P/J, I'd want it to be noiseless, so I'd probably get a good Fender like the Tony Franklin, and install Lindy Fralin pickups with the noiseless Split-Jazz at the bridge.

As it stands, I have a G&L SB-2, which is phenomenal for rock, with a very aggressive, bottom heavy split-coil tone. I don't use the J pickup much. It produces good tonal variety, but it's noisy and I prefer the 100% split-coil most of the time anyway.