This is kind of up my alley. I modded an old Squier and got rather absorbed in finding good 5-string P-bass pickups. Consequently, I've had several brands in several different basses.
A you tube search will get you some info. I recall Friedland doing a demo of a Fender P-5 which pretty well demonstrated it's sound. Played one myself. I liked it. Indeed, it sounds like a P-bass. Very classic.
Nordstrand makes a really nice P-5. In a standard 4-string size. It's vintagey - IMO the sound is toward the smoother side. Punchy and a nice P-bass sound. Clean.
Bartolini makes 2 of them, both in standard size shells. I like them both - one is Classic bass and the other is Original bass. The Classic has a good bit more high end than the Original. Both have exceptional punch and growl like a demon in a bass with swamp ash body and maple neck/FB. For my money, I liked the Original, less bright one best. Has a lot of pop like a vintage P-bass. I don't have to fight clacky high end with it. It also has killer deep low end. Playing the B-string, it makes my band's 2x10 Roland pump out air like it has fan in it.
The Yamaha is indeed good. I have a BB415 and it hangs right in there. Like it a lot. But I don't think the individual pickups are available, and they are different sizes for the bass/treble side, like the Fender.
I also tried to do the P-bass sound in a soapbar shell thing; results were less than satisfying. The EMG version was better than the Bart version, although they both sounded good for what they were.
In the end, I'm with Bart P-5s. I like the standard size better than the mixed sizes of Fender and others. The punch and growl of the Bart Original Bass P-5s are where I need to be for the funky blues that I currently play, sound kind of jumps out of them, and I really like them. These Bart Originals are in a Warmoth bass I built, swamp ash with maple neck and fingerboard. My band think it's the best sounding bass I play (compared to Pedulla, Modulus, Fender and others I own).
If you need it, make it. If you can't make it, make it better.