So I spoke to a rep at Fender today. They obviously wouldn't take responsibility for this, which I understand. What they are going to do is send me a letter on their letterhead stating that this particular bass and the serial number attached is a Jazz. Since I like the way it plays and will have this letter (in case I ever want to sell it), I will probably end up keeping it.
The more I look at it, the more I think you have a Jazz neck with a P decal. Kinda cool.