Diff issues are quite common on the Jags. The problem is that most dealer/repairer mechanics don't have the knowledge and experience to repair diffs - you need a specialist diff rebuilder.
I am an old-school mechanic and have rebuilt plenty of diffs in the past. Its a black art to get them quiet. Mostly the dealers will replace the oil seal and just torque the pinion nut up to factory spec. That is NOT the way to do it, because it takes no account of the fact that the collapsible spacer (which sits between the pinion bearings and provides preload) has already been crushed when originally tightened. Re-tightening to factory spec actually crushes the spacer a bit more and puts too much preload on the bearings. They make noise, and furthermore the gear teeth get out alignment and also make a noise. Eventually the bearings fail.
The best way to replace the seal is to use the original nut and mark it. On reassembly use so Loctite and return it to the exact same marked position. The tightening torque will be way below factory spec.
In your case the damage is already done. The diff will need stripping, new collapsible spacer, then tooth contact setting with new shims. It will be expensive (labour).
If you find a good used diff then go that route. But you won't know how good it is until its installed and been driven. You could be back to square one.
Inadequate spec on the original seals in the problem here. Bloody JLR lack of development/testing/QC !