Sorry to hear that.Looks like that is a design fault or something else,but this failure happens regularly.My friend's 08 X5 4.8i just had the same failure few days ago.
$4000 sounds absolutely outrageous.Here is a part # for a rear differential for 08 X5 4.8is Euro,RHD- 33107552530 .It is $2050 (before discounts) brand new here,so you should be able to find a used one and have it shipped for much less than that.