Hey Mate, did you eventually solve your Rear Differential problem? Most times just getting the actuator motor out from the assembly and servicing it solves the problem. You also have noticed a fault on the dash screen "TRANSMISSION FAULT, TRACTION REDUCED" immediately when you start or switch off engine.
Take motor out, hit with a wooden or rubber mallet while you try to feed with power from battery manually.
Replace the serviced motor, change the differential Castor Oil(if not replaced in a long while) and get someone with SDD or JLR IDS to reconfigure rear differential module. You should be fine if not replace Motor.
Also take note of a Fusible link fuse 40A Power supplier in the Battery Fusebox and also there is an ignition supply fuse(No 24) in the passenger side fusebox, check if they are still intact.
I have a diagram of the Electronic Rear Differential illustration attached.
Good luck
