Today I took my SX to my local music store and A/B'd it with a Fender MIM Standard Jazz 5. Played through the same amp with all amp settings at 5:00 and both pickups and the tone knob all wide open. While the Fender had a slightly cleaner tone than the SX (which could have been due to the age and type of strings, the SX had 7-month old halfwounds and the Fender had new roundwounds), there wasn't a significant enough difference between the two and the SX pickups were actually hotter.
I also asked about the cost of changing out the bridge and tuners on the SX. The cost for parts alone was $200, which is $50 more than I paid for the bass itself, not including installation costs.
Bottom line: sometimes what you have is just fine. I will keep playing my SX for now, while keeping my eye on MIM Jazz 5's on eBay and talkbass. If one comes along that's priced too good to pass up then I will pull the trigger on it.