So I'm late to the party, as iaresee has already bought a bass (congrats Ian), but for anyone else wondering which bass to go for I'll toss my opinion in there and join the choir that preaches to go with a P-bass.
They just work as Leon said, both live and in the studio, and they just sit in a track better than everything else.
Jazz basses are nice as well and have smaller necks, but I clearly prefer P-basses, I have small hands and I can make it work no problem.
I'm absolutely primarily about guitars but enjoy playing bass and have a really great SVL P-bass and a Strandberg master build P + M (Musician pu in the back+ P-bass pick up).
If I were to get a great bass on a budget around what iaresee posted, I'd look for a deal on or a used Fender Nate Mandel P-bass, it's an absolutely killer bass that punches way over it's budget IMO.
I am in the same market for a bass. These caught my eye but I have not been able to play one yet.
https://www.sandberg-guitars.de/basses/
Strandberg build very nice basses, really high quality materials, build and sound quality, and they play like butter.
I got a very good deal when I ordered, so I went in full on and got one of their vibration treated masterbuilds and it is a killer bass in every way, but every Strandberg bass that I've played have been really great.