Everyone is going to tell you their own favorite, and every P Bass is gonna be a little different (even within the same builder), and they're all P Basses.
Unfortunately, with that in mind, the best advice i can think of is just to play a lot of different P Basses and see if you fall in love with any of them. I say unfortunately because so much of the way buying instruments works these days is buying things sight-unseen online based on manufacturer/luthier reputation alone, because it's really hard to find brick-and-mortar stores where you'd be able to actually play a selection of properly set up well made basses before you buy.
Personally have never been a P-Bass person, until i walked into a local boutique/luthier shop near me and fell absolutely head-over-heels in love with their P-Bass and ordered a custom build from them a couple weeks later.
That said, having an overwhelming list from everyone's suggestions sounds like a fun research party for yourself, going to the different makers' websites and ogling the different P Basses.
Invalid Link Removed is a great option at your budget because you can pretty much design your own bass, top to bottom. You just have to build it when the parts arrive, or pay someone to build it for you.
There's also the
Maruszczyk Jake, their P-style. They have a few different sub-models of the Jake to emulate different generations of the P-Bass. Also, the Maruszczyk bass-configurator is like, dangerously fun to play with. You can design a custom P Bass for a fair bit below your max-budget, and have a bass that pulls punches way above its price-point.
