If one bass is the goal, [one bass to rule them all, and in the mix bind them] perhaps a bit of modding is the route to go. You pick the bass you like the look and feel of best, then you add the missing pickup into it.
I love my P bass, but I also love my Jaguar (Jazz-ish in flavor). My next evolution will be getting another Jag and reconfiguring the pups. I've kicked around the idea of dropping a P pup between the two J pups, or possibly changing things up completely and putting a MM pup between them (or in the bridge spot, or neck spot; the possibilities are endless). In the J-P-J configuration, you could use a simple 5-way switch to get any of the combinations you might like. To preserve a more stock look, you could get a rotary knob-type pickup selector.
All that said, I'm still keeping my P, no matter how awesome my new Jag becomes as it was my first nice bass and it sounds so good.