I don't think a P & J pickup on together will cancel too many mids, as the construction of both pickups is pretty dissimilar. I do think a more powerful J pickup has more influence over the blend as opposed to a weaker, vintage J.
Yes, there is. You, of course, will get hum. The Apollo might work fine for you, but you will have to turn the P-Bass pickup down to get a 50/50 balance if that is important to you.
The problem is that a bridge Jazz doesn't have a lot of output compared to a regular P pickup, like an SPB-1. An Apollo, which has a little more output than a regular Jazz pickup, still doesn't keep up with it. Yes, there is a tonal difference between the Apollo (more scooped) and a Hot Stack...
While I love the Apollo, I can say that if volume balance is important to you, I'd pick something slightly hotter like a Hot Stack to balance with the P.
I have played an Ernie Ball Silhouette Bass, which is sort of their version of Fender's Bass VI. I think there is an OLP version, too, which might be ok with a pickup change. :)
While I don't have a Deja Vu here, I do have a Vapor Trail, which has an insert jack. I use it with guitar, though, and I put all sorts of things in that loop, affecting the echoes only...from phaser, EQ, distortion, and even another echo.