So, I love my BB734a (active big brother of the BB434), but that doesn't mean I'm just going to blindly recommend the BB.
If you're cool with the neck, and you really want the authentic P sound, and are worried about brand name, then I would advise you to get a Fender P (or PJ, as noted, not really much difference in the P tone). The main reason I didn't get a Fender is the neck; I'm just not a fan of that chunky feel. I played a bunch of Squier and Fender Ps and PJs and just never found one that felt right in terms of the neck. (And also wow, there are some... interesting QC issues in what Fender puts out.)
The other thing to consider is if you have a strong preference on the fingerboard wood. If you want maple, then you're good to go on whatever. But if you want rosewood, then your only option is the BB, as the Fenders are all pau ferro, which I don't like, both in terms of feel and aesthetically.
Sweetwater's been great. Will definitely keep ordering from them when I can. When my BB arrived the setup was near-perfect right out of the box. The only thing I've changed is that I actually raised the action just a hair because while I'm OK with low action, well, it was actually too low (which I didn't think was possible, but there you go).
Nobody near me had a BB to try out—that's one of the reasons I bought from
Sweetwater, so I could return it easily if I didn't like it. It's not perfect—the body is obviously on the big side, mine is heavy even for a BB—but it sounds and plays
great. The neck is exactly what I wanted (kind of in between a P and a J, but a little closer to a J) and the tone is terrific. On passive, it's not a perfect duplication of a classic fender P, but it's close. In a live band context, I promise you nobody in the audience will be able to tell the difference.
And, of course, you also get the J pickup, which sounds also quite like a Fender J, though IMO not as much as it sounds like a P on the P pickup. Plus with the 734 you get the active option, which changes the sound a little plus gives you all the tonal options of having EQ. I mostly leave it in passive mode, but there are times the active is the way to go. Still in your price range, too, as these run about the same as the Player series Fenders. ($50 more, but includes a nice gig bag, so pretty much a wash.)
So, really, if you don't mind the chunky neck and don't mind not getting rosewood, I'd get the Fender. The Jaguar is more versatile so of the two Fenders that's the way I'd go, though your color choices are pretty limited.
But for me the neck was a deal-breaker. If that's where you are, you can save some money and get the 434, or spend the about same as the Fender and get the 734.
No matter what you get, if you're not buying in person I'd go with Sweetwater.
