I play RATM, RHCP and lots of other varied stuff, and since I got the Helix Stomp, I can get all those sounds out of the box without worrying too much about the actual bass.
IIRC, Warmoth do/used to do a J body with a rout for a P pickup in the right position (they also do/did the pickguard, but it's been a while since I looked), and oddly, though I don't like the look of Jazz basses at all, put a split-coil pickup on one and it's suddenly a lot more attractive. Yeah, I'm a weirdo... ๐