So, based on research so far, including many conversations with a jaguarforums.com member Cambo (who most if not all of this info is derived from), it looks like an LSD doesn't have to be terribly expensive after all. There are a couple of options besides the Quaife. I've only been researching this for the XJ (X350) cars, though S-Type info can be extrapolated.
First, a note! Changing ratios cannot be done to all of these cars, apparently the computers freak out and put the car into limp mode when they see conflicting signals about how fast different things are spinning. However, the 2006 and up has this info stored in a different file that supposedly can be changed using a dealer-quality computer (though dealers won't do it, you'll have to find an indie). I'm still researching these details.
Adding an LSD, but keeping the ratio should be straight forward as far as the car's computer goes since it simply doesn't know anything is different. It seems at this point that the pre-2006 are stuck in this boat.
That brings us back to the Quaife, the only one that is known to fit in the stock pumpkin, or switching the pumpkin and half-shafts to something that fits and has the same ratio or fitting your current stock gears into the new (used) pumpkin.
The first gen S-Type has a rear that can fit a ford 8.8 LSD unit, but not the 8.8 gears. The half-shafts will also need to be sourced. I believe the gears from the later difs will fit just fine. These 4.0 differentials and half-shafts supposedly bolt right up and accept the ring and pinion from the gen 2 allowing you to keep the ratio.
The second gen cars have the same goofy offset, unequal half shaft units as all of the x350s, so that's why 1st gen only.
Another option is the first gen Lincoln LS. It is reportedly the same size as the first gen S-Type's just doesn't have the cooling fins.
So, if you want to keep your ratio, get a quaife. If you don't want to go the quaife route, get a pumpkin and half shafts from a first gen S-Type (or try the Lincoln LS parts), swap in the gears from your car for the right ratio, and source a 8.8 LSD. Depending on how good you are at scrounging, this is significantly cheaper! Quick ebay search, it looks like about $500 for S-Type parts and a 28 spline LSD(not including any rebuild parts though, if you want to do anything 'while you're in there') the Lincoln stuff is cheaper yet.
If you have a 2006 and up, do the same, but keep the 3.31 gears in it, or change to whatever else you find that fits, then have some electrons rearranged to keep the computers happy.
This is all for informational purposes only, by all means go buy parts (and whole cars) and wrench away, just know that you are out on the front lines with me here. Cambo tells me that one of the UK forum members has added the 8.8 LSD on his S-Type, switching out the first gen S-Type's gears for the ones in his 2nd gen, to keep the computers happy.
Hope this helps!