I put EMG Geezer Butler pickups on my MIM PBass. I get compliments all the time now on how great it sounds. Definitely worth a look. 
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Y'know, I LOVE Lindy's and I recently put them back in my Glaub because the overwound SDs I had put in there were too fat, bassy, and aggressive--very much like what the OP is after I think. I wanted a more "airy" vintage tone, which I think the stock wound Lindy's provide in spades. So, for Lindy's I wouldn't get the stock wind for what the OP wants, but I'd get the +5 or +10% output option. You can email or call Lindy and tell him what you're after too if you need advice. He's a great guy, and will chat with you about it. P-Bass - Lindy Fralin PickupsLindy Fralin
Also Mike Dirnt pups sound great IMO...they are supposedly '59s. I also like Rio Grande.fender custom shop 59 pups sound huge, imo.
Have you tried rolling off the treble on your Jazz? I know the QOTSA bassist plays both p and j basses and they sound pretty consistent. Just a thought and it would save you from buying new pick ups.
Also, isn't the SP you are using a guitar pedal? You may be losing some low end with that engaged. You may consider a bass specific od/distortion. I have a DG b3k and it is extremely versatile with no loss of low end.
Good luck on your quest.
I use a maxon od9, a guitar pedal, for my fuzz on bass, and there is a definite loss of low end.
Trying a new pedal might be less hassle than soldering in new pickups.
Honestly, I've been pretty amazed at the hideous quality of pickups Fender has been using in their P's over the past few years. It's as if they are squarely going for the bedroom player market by adding a lot of treble and scooping out the mids. The basses themselves are usually quite good, but the pickups have been utter garbage as far as I'm concerned. The ones in the AmStd are better since 2012, but IMHO they still have a way to go.I would do some checking on your MIM P-Bass as P-Basses should never sound "thin". Maybe it's wired wrong or something. It's also possible that you got a lemon from a tone wood perspective. You might also want to experiment around with amplification choices and/or EQ settings. P-Basses don't have inherently super low frequency bass response. I've found that amps with their bass controls centered around 40 Hz don't work quite as well with P-Basses as amps with bass controls centered from 60 Hz - 80 Hz because that's where the actual signal from the P-Bass centers around ... but that's just my opinion ... I could be wrong. ;^)
The thing is that the P pickup, being a split coil rather than a full-width humbucker, will always have that scooped character - the coil not producing signal acts like a filter/choke on the signal being produced by the other coil.
In your position I would (and have in the past) go with the P/J option...
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I'm not quite happy with the sound of my 2014 MIM P bass. It's all stock and feels great, looks great, but since I got it I haven't added it to the live rotation because I just can't get used to it's sound. My main bass is a Jazz and while I like it there are some songs I want to sound smoother, warmer, and rounder.
I've decided the P bass is too thin sounding, not enough low end and too trebley, almost scooped sounding.
I use fuzz (swollen pickle) 75% of the time. My band is a heavy alternative three piece so standing out in the mix is paramount to me.
I've considered having a J pickup added to make it a P/J but I want to try to solve my issue without routing a new cavity.
So far I'm looking at the EMG GZR pickup, but I've never really liked EMG or signature model anything really.
The other one I'm considering is the Fender Original 62 P pickup.
I've never played either of these or heard them playing my style, (think QOTSA) so I'm not sure how to make the most educated decision.
Any suggestions or advice would be appreciated.
+1, Love mineAguilar AG 4P-60