$4000? If you have that much money and you know exactly what you want, do it yourself. Buy a custom body with your wood, weight and finish specs from MJT or some other quality builder. Find a neck you like from Fender CS, Warmoth or another reliable source. Get fantastic pickups from any boutique shop: Lollar, Fralin, Nordstrand, etc (or have them hand-wound). High-quality knobs, tuners, bridges and other parts are readily available online. Buy a pre-wired harness with great components so you can wire it yourself, or have the whole thing assembled by your trusted luthier. It's just a P bass, not an exotic coffee table neck-through with active pickups and a control panel like a spaceship. You should be able to get
exactly what you want for less than half your budget.
I built a J bass customized to my specs and the whole thing was less than $1500:
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