I can't claim to be ready to take the LA session scene by storm, but in a previous life (25 years ago), I was a serious (way too serious) composition/theory/classical guitar student. I went as far as a Masters degree, and if I had completed my dissertation I would have a Phd.
So now as an aspiring funkateer, what of all of this classical training do I find direcly applicable? Only the basic stuff. I can read bass and treble clef, have no problem spelling chords and find them on the piano, understand the power of the circle of fifths. On the theory side that just about does it.
Classical tonal theory uses some of the same terminology as jazz theory a la Mark Levine, but there is significantly less emphasis on modes. Classical music took a wrong turn somewhere around 1800. Your standard classical harmony course focuses a lot on what is called 'figured bass'. Back in Bach's day, organ players would get a bass line with notations that indicated what intervals should be present above, and they would just improvise within those constraints. Only hard core early music specialists do this these days. Everybody else learns the rules so that they can put together 'correct' 4 part harmonizations, and in most cases (mine included) this is done mechanically, not by ear. Still you do learn about key relationships which does help when trying to put together bass lines. As a composer, I was into atonal stuff, including 12 tone theory, which so far does not seem to be particularly relevant to non classical music, except perhaps the most avant-garde jazz. 12 tone theory is radically different from tonal theory in that it is based on order not content. In tonal music, you pick 7 notes around the circle of fifths and you get a diatonic collection (a k a key). Chromaticism is understood in term of passing notes between scale degrees and membership in a particular key (e.g. if you are in C major, and a Bb appears, you think heading toward F major). 12 tone music keeps all 12 pitch classes in play at all times, and it is only the order that they appear that makes a difference. The result tends to be pretty dissonant, and intellectual. If you are curious, I would direct you to Donald Martino's wonderful chamber piece Notturno, which is my favorite in this genre.
My classical guitar training is more helpful. All of the left hand stuff transfers pretty directly to the electric bass. More of my right hand technique should be applicable, but I stupidly put my blinders on and have been using i and m exclusively for finger style, even though I spent years developing an articulate thumb and ring finger. I intend to open up my right hand technique and find more use for these fingers. I saw Charlie Hunter last night for the first time, and he definitely inspired me.
I am rambling. I am also someone who decided to find a different career 25 years ago, so YMMV. If the classical theory stuff is approached as ear training (which almost nobody really does), I think it could be liberating and helpful. Most popular music seems go by in 4 and 8 measure chunks, while if you study the great German composers, you see a lot more phrase length variety.
In closing, I will confess that one area where my classical guitar training really hurts me is my time. Because you are playing solo, and the instrument has limited dynamic range, a lot of expression is done via rubato, which is death to laying down a solid groove. I spend a lot of time with a metronome atoning for this sin.