The P-Bass is widely considered to be the benchmark of bass guitar tone; definitely a solid instrument. Is it a 4 or 5?
Well it depends for what type of music...I've seen more Jazz basses out there, especially in gospel, r&b, jazz, etc....The P-bass was definitely the first of Leo Fender's revolutionary designs, and was probably the first popular electric bass (had to do report on Fender basses, don't mind me lol).
P-bass 5 strings, ESPECIALLY Fenders, are commonly lamented for their B strings....I'd definitely play one before buying. You might like how they sound. P-basses generally have one main "tone." I wouldn't say they're very versatile. Jazz basses are way more versatile and sound better IMO. See if they'll give you an American standard J-bass for the same price.....
Thaddues Tribbett plays a P-bass....I don't think it's a Fender though, so you can definitely get a nice tone out of one....
I've bought/sold/traded a bunch of guitars on Talkbass....with your budget, you can probably find a used $1500 bass on there. Generally people have good reputations on TB and are honest, I've only ever had one bad transaction, and it really wasn't the other party's fault.
Again, I'd suggest the Jazz, or the Brute lol. My opinion of course.