The LP Sig was actually conceived by Les Paul to sound like a P Bass (hence the long scale which he preferred on a bass) because he liked how P Basses sounded, he just wanted a bass that could do a few more sounds in addition. If you put an LP Sig varitone in the position that is the least loud (excepting that control board tone you get from the bottom jacket), then it has that sparkle of a good P Bass. Using the other positions "gibsonizes" the sound more and more.
Now, for whatever reason, they did not replicate that with the Epi JC, that Electra pup is thuddy no matter what position you use, in that sense the JS is more faithful to what you would expect from a Gibson bass, but not faithful to the original intents of the LP Sig. I don't think it's an issue with the construction of the bass, it's the pup and electronics. I'm sure Rob's JC sounds a lot more P-ish now with the Triumph pup.
My Fender playing buddy here exclaimed first time he played the LP Sig: This sounds like a good P Bass! He's never said that about any other of my Gibson basses.
Uwe