"Chain replacement"??????

That's a new one.

Yeah, maybe the transfer case actuator gear failed instead. Search for that replacement procedure, it's cheaper and might be the cause of your issue.

Found a video on Youtube for the E53 TC rebuild,
https://www.youtube.com/watch?v=FbZmmOphYw0&t=163s
The ATC700 for the E70 is not much different here is the list of changes
Transfer Box ATC700
The transfer box ATC700 is a further development of the ATC500 that was fitted in the
E53. The power is output via a chain to the propeller shaft to the front axle and apportioned
by a multi-disc clutch.
The torque transmission efficiency of the multi-disc package has remained the same
compared to the ATC500.
Only slight changes have been made compared to the ATC500.
• Installation position of coding resistor
• Optimized lifetime gear oil
• New ventilation system
• Transfer box control unit (VGSG).

Here is another video of interest

https://www.youtube.com/watch?v=TivwD8MTMWg&t=12s

Why do you think the chain has failed? Most likely cause of failure for these is the little plastic gear in the controller. This can be replaced for about $100; (seems like a lot for a little gear, but this is an aftermarket solution, and they clearly intend to make a good profit....but still way better than replacing the controller or the whole TC.)

Transfer case fluid change and adaptation cycle
 

I am happy to report that the problem got fixed with the transfer case fluid change that I did and took it to my local BMW mechanic who ran the adaptive cycle and the car drives much better and no shudders in slow tight turns under power. The plastic actuator gear may wear out sometime in the future, however, I can handle that myself. Thanks for all the information on the forum, it was a great help, thanks guys.

BMW suggested TC replacement for 5-6K, got lucky with the TC, changed the fluid and had a BMW shop run the adaptive cycle for the TC and the problem went away, no shudder, car runs like new. Have not changed the heater plugs yet.

Changed the fluid and had a shop run the adaptation cycle for the transfer case, problem solved