Best budget jazz bass I've had my hands on was a Farida FJB20 (I believe Farida were Dawson's music own brand, sunburst, maple neck with block inlays). Going back a few years but I got to spend some quality time with one as the owner had abused it a bit and dropped/threw it down face first and knocked 2 of the tuners out of the headstock, stripping the screw holes. Wanted that fixed and pickups upgraded at the same time. If you could find one of those I'd heartily recommend it. It didn't need the pickups upgrading to my ears...
Beyond that it'd be Sire every day of the week for me on that budget. I haven't tried the v2 versions with the newer "worn in" neck shape, but I can only imagine how nice the neck feels having loved the 5 string V7 I had for a while.