Having copied his tone quite a few times unabashedly in my younger days, the recipe is pretty simple, but there aren't many substitutes.
1. PJ Bass of some sort (A jazz bass with 70s position bridge pickup like a Geddy Lee can get in a rough ballpark, but only if a PJ isn't an option. They're bit twangier than a standard jazz and slightly different mids)
2. Fresh strings (preferably steel strings like Rotosound 66's. I've had great luck with DR Hi Beams and they last much longer)
3. GK amp. There's something about the GK tone stack and baked in eq that works with his tone. You can dial out the baked in tone, but it's hard to add it to a non-GK amp. I've used everything from the 400RB to MB Fusion and they all work.
Supplemental: Chorus pedal (really any decent one will do) for a few songs, and maybe a bit of compression for the pick attack. I wouldn't worry about any sansamp except maybe to tweak the eq a little or add a tiny bit of grit. The GK is the predominant tone factor amp-wise for his sound. The RBI is a very different tone and investing in a rack unit is dumb for a limited application. If you absolutely want one, I'd suggest a para-driver di or a Q-Strip. The para driver is much more versatile than the RBI/Bass Driver (it's the RPM rack in a box) and the Q-Strip is a wonderful eq/tone sculpting tool.
My advise is to buy a PJ bass of some sort. Swapping pickups on that jazz will be a money pit and won't get you closer to his tone at all. It's about the pickup configuration and placement. My personal recommendation is a used Japanese made Fender Jazz Bass Special from 1984-87. They are incredible versatile basses, lightweight, and comfortable. If you can't find one for a decent price, Fender just reissued a fairly close copy of them called the "Boxer" bass. Having played one this week, they nail the tone including the TBX control and don't have the painted black neck of the originals and his signature bass. Also, get a GK amp of some sort. If you can afford it, the new Legacy amps are fantastic and sound much like the old RB series while being lightweight and having modern features. If you can't, a used MB series head or combo (including the fusions) will do the trick as will an old RB series if you can heft the extra weight.