Appreciate the input and perspective, thank you for that.As an owner of a couple of Mexican Player Ps, I caution you about chasing something perceived as better when, in actuality, the stock electronics sound very freaking good on these basses.
The Alnico 5 pickups used in these production models are a little bright. With the tone fully open, it is a little too bright, making them sound thin. As soon as you roll down your tone control, you get a very warm, round, the classic precision bass sound we all fell in love with...
...The thing about the stock pickups is, you don't realize how good they are until you hear how they sound with a rhythm section. The reason Pbasses sit so well in a mix is that the midrange response is perfect. It does not overwhelm the kick, and its not to sparkly too compete with cymbals and guitars. The most important thing about the sound is how well it works in the ensemble setting, not how it sounds soloed in your practice room. FWIW
Record your rehearsals, shows... You will be very surprised at what you have.
For the sake of discussion, as I stated earlier I've been playing this bass in both group rehearsal and live as well with some regularity for over 3 years now, so it's not like I'm jumping to conclusions or haven't tried a lot of different amp settings, different string brands/types etc. Generally I do keep the tone all off or perhaps up just a small bit due to the tone knob not really giving me much range tonally, just more thin brightness. Agreed, they can/do sound a bit thin and bright with almost any of the tone knob engaged and I do/have listened to both live room and board recordings as well as what I'm hearing in my IEM's, so I would like to believe that I have a fair idea of what's going on with it. And it sounds "OK" for the most part. I believe what I'm ultimately looking for is just a bit more definition in the lower end (the undefinable "punch") without getting muddy. I get that from the Sadowski's without any issues, I just want/need to find out how to get it from the P as well.
I'm playing with this suggestion as well (thank you @MartinB ) and trying a fresh set of strings that I hope will get me a little closer. Honestly it's a quality instrument and I don't think the bass in and of itself is the problem as much as just finding that magic combination, which is what drove me to talk about the potential of changing up something electronically initially. I'm convinced it's findable, just gotta keep digging...The Sadowsky preamp pedal manual says it boosts at 40Hz and 4kHz. Since you've already got an HX Stomp, you could try adding an EQ block (either Low/High Shelf or Parametric) at the start of the chain to boost those frequencies and see if that gets you closer to the sound you want.