As others have pointed out, a PJ won't sound like a P. The reason is that all components in a circuit interact. The only way around that is to have a cut-out switch that disconnects the J pickup so only the P (split coil) feeds the jack.
There's a whole encyclopedia of tweaks one can do with complex volume and tone circuits.
One video of a Peavey T40 owner, setting it up to emulate other basses, demonstrates some of the quirks of components that affect the overall tone, even when dialed down. (T40s had a coil blend tone control that gradually changed a pickup from single coil to dual coil. Lots of tweaks in that, for sure.)
There's a whole encyclopedia of tweaks one can do with complex volume and tone circuits.
One video of a Peavey T40 owner, setting it up to emulate other basses, demonstrates some of the quirks of components that affect the overall tone, even when dialed down. (T40s had a coil blend tone control that gradually changed a pickup from single coil to dual coil. Lots of tweaks in that, for sure.)