The reason why I asked if you had bottomed out at some time (pot hole, road hazard, whatever) is to suggest that the driveshaft may be out of alignment, although this seems remote. Similarly, I was going to suggest that a rim could have gotten dented, but as you recently put on new tires, I should think they would have caught something like that, though not necessarily.
I think I came across somewhere that "M" cars do take a different differential fluid, but why that should apply to a non-M doesn't make sense to me. But that does raise a question, any fluid leaks in the rear end, or elsewhere? Torn half shaft boots?
By the way, is this an automatic or a manual?