Ok...if it was me with your budget AND I was unable to find a Fender with the specifications you've set out, I'd find a US/MIJ P/J pickup'd Precision and source a Jazz bass neck.
All the Fender stuff for 34" scale basses is pretty much interchangeable, even down to the screw-holes, so old neck off, new neck on. Find a local guy to do a decent fret level and set up and you're ready to go.
In a blindfold test, I doubt many would be able to tell the difference between a Mexican, Japanese or US model, although this is speculative!