So I'm going to be taking my MIM Jazz Bass V into a shop next week to get some new strings on but also the action set. I'm really after string action similar to that found on Foderas and the Billy Sheehan signature Yamaha (ridiculously low). Will my bass be able to do this or would I need to consider other options such as a better bridge (if so what type?), or possibly getting the current bridge counter-sunk?
The Relief adjustment of the neck, the quality of the fretwork and how soft/hard you play will dictate how low the action can be set. Unless there is some inherent problem with the bridge, which is unlikely, it won't be the issue.
Having a good fret level and dress along with a new properly cut nut will do a whole lot more changing the bridge.
The key thing to having really low action is in the frets. They have to be very precisely leveled. If they aren't, you can't adjust the action low without random buzzing problems. Forget other bridges and things. The fretwork has to be right first. The Tech you take it to will first try to set the relief and action as low as it can go. Then you'll find out what condition your fretwork is in. If it's not low enough for you, he's probably going to recommend a fret leveling job. The cost of that will depend on how far you want to go. That's what you can expect.
Try to get it right by just adjusting the truss rod first. Depending on what kind of bow you have(most headstock adjustment truss rods tend to start bowing at the 15th fret), you may have to take it slow and adjust in increments every day until it's right, or you'll have a neck with multiple bowing issues..which can also be fixed. Get it to where it buzzes, then raise the saddles a little. If you can't get the low action you want from these options, take it to a tech for fret work. *There's also the shim method. Check that out too.