Below is my perspective of your opions. If you could be so kind, please think about the following question because they will influence what option you will want to persue.
I am sure you are aware that a 2006 STR with have the Gen II compact modified 8.0 differential, where there never has been an available off-the-shelf LSD.
Questions:
1. What do you use your STR for? Daily driver? Weekend driver? Track car? Autocross? Performance driving? - High speed straight line acceleration or perhaps twisty winding road driving.
2. How much do you want to spend? (Budget?)
3. How long can the car be out of service in order to make changes to the rear end?
4. Can you do larger more complex car projects by yourself, or do you have to pay a mechanic to do the work?

Options: (USA Dollar estimate)
1 Rebuild the OEM Gen II differential - bearings and seals are still available,I would recommend installing a new pinion and ring gear (if available) This is the most cost effective way to keep the car running, and a good rebuild should be good for anpothe 100K miles. (Under $1000USD parts, probably $600 to $800 labor. Car out of service for 3 to 4 days assuming you purchase and receive the rebuild parts before you start.
I recommend this approach if your car is a daily driver or weekend driver, and improving the car's handling and performance is not required.

2. If you convince yourself that you really need LSD. Purchase a rear sub frame with the differential and axles from a 2003-2004 STR - Needs electric parking brake compatability), (Gen 1 differential came with equal length axles.) (Your driveshaft should work)
Have the gen 1 differential rebuilt, and install a Quaife LDS. (If you can find one). After the spare rear sub-frame, differential and axles are prepared to your level of detailing, drop the rear subframe assembly out of your STR and swap all the suspension parts . (You really should determine, ahead of time, if any of your suspension bushings need replacing, and do that while you are at it.)(Also may want to replace the rear hub carrier bearings if they have higher miles on the car.) Put the rebuilt rear sub-frame assembly back into the car and hook everything back up and bleed the brakes. If you can get all of the parts ahead of time, one can do the subframe swap in a couple days.(probably 1 to 3 months in preparation - purchasing parts, having the differential rebuilt...) Every detail you forget to think about will keep the car out of service for more time. Labor will probably be in the $2,000USD to $3,500USD range. Parts - Sub frame with differential and axles, $600 to 800 + shipping, Quaife LSD $1100 to $1300, rebuild kit with ring and pinion, $600 to $800. (Suspension parts to be replaced, $300 to $1000. (Probably should budget up to $5K USD to have someone do this for you.)

3. As had been discussed in this thread, Third approach is for the most dedicated enthusiasts, who wants/needs a heavier duty differential and a Torsion T2 or T2R, or Eaton Trutrac torsion LSD differential.
This conversion will provide LSD capability that one will like for autocrossing and tracking your STR. Like option #2, you should purchase another sub frame, weld on mounts for a Cobra/Explorer IRS pumpkin, decide on 28 or 31 splines axle hubs,have the Torsen internals or other LSD installed, have new axles custom fabricated/assembled,custom driveshaft and completely build up a rebuilt subframe assembly. As and example, the one I build up cost me close to $5,000 in parts, and many hours of labor.

4. Be a pioneer and install a later model year Ford or Jaguar IRS differential. I've researched this to some degree, and it would be a fun project to execute. Some day...

Edited: I looked up the current cost of the Quaife ATB Differential - QDF3W Street price apperas to be around $1.500+-. Wow, thats expensive. I think I can purchase a Torsen T2R for around $800, and a Cobra IRS pumplin for around $300. Pinion are ring would be the same for any rebuild. If one can figure out a cost effective axle conversion, one might come out equal price wise swapping to the Ford IRS differential.

 

In as a few words as possible, what are the issues with putting an 8.8 lsd carrier in the gen 1 8.0 pumpkin and then bolting in the entire gen 1 rear subframe? let’s just forget about the epb

 

Tijoe has laid it out above your post the best I have ever seen. It's still fairly complicated. You need to read and understand all the differences between generations of the S-Type and the Lincoln LS.
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Axles, and ABS tone rings Gen II axles are different lengths, Gen I axles are the same length. Gen I used a press on metal cut gear ring tones, Gen II uses metal band style ring tone. (Don't remember if they are the same tooth count. But the ABS system changed between face lift cars.)
Regarding the Drive shaft, I believe that Gen I and Gen II used different rubber couplers. P/Ns are different. (I didn't pay much attendion to this because I had a custom drive shaft fabricated that used a regular heavy duty U joint.) But, the interface plane is the same between Gen I and Gen II drive shafts.,

Therefore you might have to make custom/ mix and match axles and hubs in order make it all work.

Packed away somewhere, I have 3 different versions of Jaguar Gen I 1/2 shaft axles, and 4 different versions of Gen II axles. They varied a lot over the years (Probably based on 3.0 vs 4.2 VS STRs and that combined with what shafts were supplied by their OEM part suppliers. I saw a lot of different axle shaft diameters and spline counts and bearing counts inside the the CV joint.

And a word to the wise: IF you go down the path of trying to make your own hybrid axles, I found out that trying to remove the axles from the CV joints/hubs was easy, or dam near impossible. (I had used axles literally explode trying to press them apart.) Spent weeks contacting 1/2 shaft rebuilders, and they said that many of the axles become frozen together and can't be presed apart and they go into the recycle bin.

 

I have a 2000 manual V6 S-Type with VIN before M45254, so I should have the gen 1 diff with gen 1 axles and after reading 400 messages lasting 6 years I believe I can just buy any 8.8 Ford LSD, it should just bolt into my housing and I just need to be careful about the spline count to be the correct one. In theory I should also be able to change the gear ratio since the manual shouldn't have limp mode issues.

Did I get lucky on my Jag model or am I missing something and it's actually more complicated than this?

Thanks

 

Read this thread in the Lincoln LS forum. https://www.lincolnvscadillac.com/fo...bolt-in.39863/
Our Gen1 cases are called an 8.0 differential. You can use a 28 spline 8.8 LSD module, but you have to use stock Jaguar/Ford 8.0 ring and pinion gears.