I think that price range (give or take) is a sweet spot for a very nice bass guitar. For production stuff, yes, this is a great place to be.
That budget opens up a lot of possibilities to find very cool vintage stuff, new high end production stuff, and some very nice custom / small build stuff.
I think you open things up even more if you are willing to go used - my only advice there is play it first, be sure, and don't get starstruck by a cool vintage bass or whatever. TAKE YOUR TIME. Know what you want, and wait for it, and be ready.
My experience is that about 10-15 years ago I was able to buy a used Fender CS NOS '64 Jazz bass for $2K. That was a LOT for me at the time. It had some marks in the finish - now it's yellowed, has chips and even more marks in the finish. I've done hundreds and hundreds of gigs with it, and made a few records too. It is an exceptional bass and a work horse.
On the P bass side of things, I've bought more than a few in all kinds of price ranges. I did buy a AVRI, which was super nice, but the 60s P neck didn't work for me. The many Ps that I bought in the cheeper price range just didn't speak to me or play that well. Just -meh. I was finally able to save some $$ and found the sweetest P. Almost exact specs of what I wanted and it is just an incredible instrument. Built custom for another dude by Brady Muckelroy. I was able to scoop it up for under your budget, and feel like I got the deal of a lifetime on a one of a kind. I honestly don't know how I could get a better instrument...
OP, you are going to be able to get a fabulous musical instrument, and IMO, something that punches above it's weight... especially if you wait for it to come to you and don't impulse buy.