I want to give it a more balanced sound between a J-Bass and a P-Bass, is there any way to do this? I've thought about replacing one of the pickups with a precision PU and keeping the other, is that a decent way?
Well, doing the routing for that will suck, if you feel confident doing that, then by all means replace the neck pup with a precision.

You might also look at some stacked dual coil pups that fit in a jazz space, you might get a slightly chunckier, mid-ier sound from those, dimarzio makes some and I'm sure other companies do as well. Nothing but a P-bass pup sounds like a p-bass pup though.