Go with the P-Bass. I own both American Jazz and Custom P-Bass. Custom P just blows away the Jazz on the neck growl and warmth. I actually prefer the more staight forward shape of the P bass too. I ended up installing a J bridge pu on the P bass so its a PJ bass now.
I just played a Fender Highway 1 Jazz the other day and would be a great addition to the P bass if you really desire the J-bridge sound.
I just played a Fender Highway 1 Jazz the other day and would be a great addition to the P bass if you really desire the J-bridge sound.