I'm a guitar player (the other kind

). I'd like to take up learning bass. I'm confident I want a Fender P-Bass to start. I've already gone to multiple stores trying them out to get a feel. I've also been lurking the forum while doing my own research.
I've been looking at the Fender Player Precision and American Professional Precision. As someone starting out, should I wait until I'm farther in the journey before considering the more expensive basses? I can afford an American Professional but I'm not sure if I should dive in like that, yet. The American Professional was my favorite of the ones I tried. The Fender Player I tried was a close second.
I'm curious what the more established bass players think.