Interesting. I love that Jaco-soloed-bridge Jazz tone. It is odd to think that this couldn't be done successfully with a PJ setup. However, ...
I'd like a P-bass but with 2 jazz pickups and no Precision pickup. Why? Well, I have several Ps and love them. I want a 60/61 Jazz bass but these, for some reason, seem to be quite expensive. I have considered "adjusting" a cheap pre-CBS Precision - obviously, one that's been messed about - and slotting in a couple of bare-knuckle Jazz PUs along with 2 tone and 2 volume pots a la 1960/61 Jazzes. This then enables a full range of tones from a lightweight and compact "Jazz", ie a Precision! Then you choose between the P-bass or the J-ish-bass.
Start the slaying ...
Peace
Davo