I've had two different (self routed) P+MM basses and one PJ.
In my experience, the PJ was almost like jazz bass with some extra balls. Stock pickups with no upgrades.
My first P+MM was a jazz bass with the humbucker in the jazz position. I should have never messed with it because the tone was great, but I really didn't care fore the body style. It had SPB-2 and a SMB-4. The sound was tight and growly with some good thump in the lower end. It's couldn't do a great impression of either the P or the MM, but together they sounded fantastic.
I currently run P+MM in a Player II. The MM is routed directly behind the P pickups in the natural MM position. I tried the stock P pickup and it wasn't even close to keeping up with the MM. Replaced it with a SPB-3 and it's definitely better, though the MM in its natural position still really covers more of the P (even if you try to really adjust pickup heights and such). I'd describe this one as tight with some wolfiness, I guess? Mostly MM sound with a little extra on the low end.
The tone is good, but if I were doing the same thing, I'd get another P contour body and put the MM back where the J pickup sits. It's also slightly unfair to compare because I didn't wire a tone knob and used 500k pots, so there's a lot of taming on the amp side. I might try to modify with a side mount jack and a tone knob to see how it feels.