I'm not sure you'll get a MIJ 62 reissue within budget, if you see one at that price then snap it up. They're a fantastic bit of kit and my favourite Fenders. If you haven't played on before just be aware that they have a gloss neck, narrow frets and a more curved fingerboard radius. Some people don't like a gloss neck, but that and the frets are what I really love about that range.
Sire do make great J's and they're worth a look. If it were me though I'd be picking up a G&L Tribute JB. The neck is glossed and it's also a tad wider/rounder in the hand (back to front, the nut width is the same) than the current Fender MiM line up. They're nice basses for not a lot of money. There's one in the classifieds now (that has nowt to do with me)