Time for closing update! 🙂
Upon disassembly, I confirmed that the insert nut was loose, allowing for excessive play in the driveline any time car transitioned from acceleration to deceleration.
Here is the visual - you can clearly see the nut was not tightly attached to the diff, causing all kinds of play. No surprise there would be a lot of vibration at higher speeds.
https://youtu.be/QaY4hkuMLh8
So, how we fix that?
Order of operations was pretty simple and ended up costing about $70 in parts (new insert nut, new guibo, 6 guibo bolts/nuts).
1) Safely jack up the car, remove exhaust and heat shields.
2) Loosen up the insert nut (it had play, BUT it was really really tight!) With 50mm wrench.
3) Loosen up center support bearing and guibo.
4) remove driveshaft.
5) clean everything up.
6) Put new guibo (not required, but I replaced original aluminum one with the revised rubber one as it supposed to be better - and it did add smoothness to when I shift gears).
7) Lube and torque as needed following the BMW manual ( start with insert nut, then guibo, then CSB).
8) Enjoy the smooth ride!
Some pictures attached for your viewing pleasure.
I put about 700 miles since the work was done - The car never felt this tight and organized.
Upon further review, the threads on the driveline had two types of locktite - I suspect one was from the factory and 2nd from when dealer installed the LSD (aka dealer did not clean up original sealant).
Dealer did likely follow the torque spec to install the LSD, but the nut reached the torque spec before it fully secured the driveshaft.
Work took about 4 hours and was a rather satisfying process as things came together quite well - especially considering how many parts needed to be removed.
P.S. - I did first try to tighten the original nut to test if this would fix the issue. Indeed, it fixed it, allowing me to go full confidence on making a proper repair.