You can do it, it's been done before (it's also been an obsession of mine for the same reasons). The simplest way to do it would be to wire all pickups through their own tone pots, join the leads in parallel and then run through a single tone pot and out to the jack.
However, there's a caveat: with a P-bass you won't get the same pickup positions as Stu Hamm. In fact, the neck J and the neck P on the Precisions and Jazzes overlap - and the position is what gives the pickups their tone. Stu Hamm started from a Jazz bass and has the middle-P specially wound in order to compensate for the fact the P pickup is moved towards the bridge.
BUT... above all, it's your bass and you might be pleasantly surprised by the result. To help you out, here's a few links:
Creating a J-P-J configuration JPJ config on Jazz Bass?