I own a Stingray 5 HH and a USA Precision 5 and have owned a different Stingray 5 and a 55-02 (and would love to try an L-2500 some day). They're all pretty different basses and I've really liked them all. I'm currently sticking with the SR5 with rounds and the P5 with flats and feel like those two will cover as much tonal range as I'll need.
My short and sweet answer - go with the 55-02 if you're comfortable with a 35" scale and wide spacing I loved mine and sold it only because it wasn't comfortable for me to play. A lot of people will say "You'll get used to it, I did, just play it and it and you'll stop noticing the difference". Wrong!!! Some people might not notice but others will and after working at it for a year and a half I had to admit it I was one of those unfortunate people. That was a bitter pill to swallow because I think it sounded the best of the basses I've owned (it had the Bartolini electronics, the LH-3s were just coming out when I sold it) and it was easily the most versatile.
If you want tonal versatility and prefer 34" scale and narrower spacing then I'd say the Stingray 5 is the best bet out of the ones I've owned. I've never had an L-2500 but I'm guessing it would also be different but also very versatile.
If you really love the Precision you have and you want to keep that sound, then the Precision V is a great bass as well. It doesn't get much simpler than that but it works so well in so many situations that you really can't go wrong with it. I've only had mine a week and a half but I can't stop playing it. In fact, I haven't even touched my Stingray since I got it (though I'm sure that will change opnce the honeymoon is over)
One other thing, if haven't already done it, considered buying used and look in the For Sale forums here. I've bought several basses here, have gotten really good deals on all of them, and with one exception they've been in absolutely phenomenal condition. Any of the ones you've asked about could be bought for half the price of your budget, so you could either save yourself $1000 or buy two great basses for that kind of money.