I have pondered this. I'm a J guy, but did have a SCPB a few years ago. The memory is a little fuzzy, but the SCPB did have a unique sound to it vs the soloed J neck PU. EQ aside, the SCPB was more thumpy and punchy. I think that both PU designs can wound and designed to be fairly wide range, but given the general winding amount and bobbin dimensions, they will have tendencies to a certain tonality. It seems the SCP PU design tends to favor a touch more mids and highs given that it's traditionally located about 1/4-1/2" further from the bridge vs a J neck PU. The low end is enhanced by it's harmonic placement in the scale. The narrow and long J PUs tend to be a touch more scooped and articulate vs a tall and wide SCP being more punchy and aggressive.
Another big difference is the J PU has the two pole per string aperture vs the P with one pole per string. This makes a very different initial attack on the note, the single pole PU will initially spike harder and decay quicker vs the J type PU's smoother attack and slightly longer decay. Seems that both, being single coils, can have a similar top end, but the J will be more articulate and the SCP more punchy. Another thing to consider is that SCP PUs have pole piece spacing to accommodate the neck position and the spacing may be too narrow for the bridge position. If there is misalignment, the string will go in and out of the magnetic field and kind of warble. Bigger pole pieces help negate that effect.
This is just my 2 cents, but given the amount of choices out there, which is like 10 J designs to 1 SCP, you may have a better chance finding a J PU that will meet your needs vs a SCP.