Your J probably sounds lifeless next to your P because: 1) standard J pickups are weak compared to standard P pickups, 2) your P has a 1/4 Lb. pickup, and 3) your P has rounds on it. Duller sounding strings on a bass with an inherently weaker pickup design, compared to more aggressive sounding strings, on a bass with a souped up version of a pickup design that is already inherently beefy. Both basses are set up to be tonal opposites of each other.
Have you ever tried the Fat Beams? Hi Beams are next on my list.
Did you sell a kidney to be able to afford those strings?I just bought my first bass a couple of days ago for home recording. I'm replacing the strings that came on the guitar with a set of Thomastik-Infeld JF344's. These strings are highly recommended.