Great video! I enjoy all of them and give you a lot of credit for taking the time to share with the rest of us.
As far as what the best gear oil is for our HAG 220 Electrically Activated M Lock rear axle final drive, (basically an electronic limited slip differential, or ELSD,) these are the four best fluids that I have found that can be had relatively easily here in the US, at least.
I believe the fluid provided in BMW p/n 83222357992 is the same Castrol fluid below, but am not certain. Bottom line is, this one isn't that hard - just pick any one of the following four and rest easy; you only need 1 to 1.5L at most to drain and refill.
Castrol Syntrax Limited Slip 75w-140 GL 5- Factory fill, formal BMW approval, (as well as Nissan GTR)
- Full Synthetic
- Made in Germany
- Can be expensive in 500 mL bottles but I found it on Amazon in 1L bottle for $29 shipped
Ravenol Getriebeoel SLS SAE 75W-140 GL 5 LS- Polyalphaolefin (PAO) based full synthetic
- Particularly suited for applications requiring "limited slip" (LS) properties
- Made in Germany
- Can be had at Blauparts.com out of Wisconsin for about $24 for 1L
Red Line 75w-140 GL-5- Full synthetic formula created from polyol ester base stock
- Made in California
- Easily had for about $18-$22 per 1 quart, (ECS Tuning, Jegs, Summit, Bimmerworld, Amazon)
Motul Gear Competition 75w-140- 100% Synthetic from Ester base stock
- Made in France
- ECS Tuning for about $32 / 1 L