Before you start looking to make changes, determine what is wrong with your current bass that you'd like to alter. That way you can make upgrades instead of just changes. Now my experience is that the MIM Standard series is pretty good except for horrible inconsistency in fit and finish, plus the pots and jacks are junk. So, I'd start with getting a really good set up, which will most likely include getting the nut cut correctly (almost no mass-production instrument has the nut really cut well, and most are just barely passable) and likely getting the frets dressed. This will aid in getting the bass to play as in-tune as possible all over the neck, and should allow you to get the action as low as you want and still have a reasonable dynamic range available.
After that it comes down to what YOU want when you play it. Bridges? The stock bridge is great unless you want more sustain or you want the sound of a high-mass bridge. The BA-II bridge is NOT inherently better in any way, and to many people it's worse than the stock bridge. The stock tuners aren't the greatest, but if they allow you to get into tune easily without having to tweak back and forth a lot (and most tuning issues are the nut, the strings, and how they're installed rather than the machines) then don't waste your money on changing them.
The stock pickup? Could be great, could be crap. But even if you get a dozen suggestions, none of them mean squat unless you can clearly identify what it is about the stock PUP that isn't working for you- and you're able to make sure it's the PUP that's causing those issues. So, my advice is to LISTEN closely to what YOUR bass sounds like unplugged. If it's bright and twangy with an ugly midrange honk you don't like unplugged, don't waste any money trying to get rid of that with electronics. You'll wind up with a crap load of money invested in a MIM P Bass that's worth the same money on the used market as an unmodded one. You'll simply have to narrow down what you're looking for here.
IF you do decide to put a different PUP in, that's a good time to replace the generally cheesy pots and the minimally adequate jack they use. Spend a little bit on good full-sized CTS pots and a real Switchcraft 1/4" jack. If it were mine, I'd do that even if I don't change PUPs.
But here's the best upgrade you can make with that bass. Invest the money in either (or both) of two things. First is lessons- as my friend Andy says "Less E-Bay, more Mel Bay". Getting a good solid grounding in basic harmony theory will do more for your bass playing than five new basses. Work on timing, harmony, grooves, understanding the language of music, and ear training.
The second suggestion is to keep playing the bass you have, and save the money you're thinking of dropping on changes ('cause it's only an "upgrade" if it IMPROVES things, a mere change is just a change) for a better bass. There IS a lot of the bass's sound and playability that's a function of the woods used and the care with which it's put together. I used to manage a store and I took a lot of peoples' money as they tried to turn sows' ears into silk purses. If the body and neck aren't well-made, stable, and well assembled, then it's ultimately a gamble at best and a waste most often to try to make big changes. I learned this the hard way myself with two basses (a '73 Precision I bought in '76 and a MIJ Fretless I got in '88).
John