Unplug the vtg motor on the back of your transfer case. Drive it. Did it go away? If so it needs a transfer case. They are common causes of what you are experiencing.
Just saw you pulled the fuse. Check all mounts and If the axles look good sounds like a transfer case.
BMW master tech
2010 E92 335i 6-speed 25k miles. - For sale.
1989 E30 325i coupe 5-speed DD