Transfer Case probs due to mismatched tires?

[MileHiX5]

I'm a newbie to the X5 family...
I have just purchased a 2001 X5 4.4 sport on 9/10...

Here's what's happened meanwhile...

The dealer agreed to look into/fix whatever was wrong due to the brake light that was on, and also install two new rear tires, before I was to pick up the car on 9/13 (after getting detailed).
The dealer was ordering 255s for the rear. I called him to let him know that 285s are on the rear. He corrected the order (supposedly) and learned that they are staggered (from me).
When I picked up the car on 9/13, I went to point out to him how the front/rears are different. Well, while Brakes Plus took care of replacing the front brakes, Brakes Plus had installed the tires back on with the 285s on the right side (front and rear), and the 255s on the left side (front and rear). Yes, doesn't make sense to me either why.....

I was told the new 285s would be several days to arrive, and the tires could get rotated correctly then at the tire shop.
When I was told that the tires had arrived at Discount Tire...I immediately took the car over for tire replacement and correction. The 255s had come in, when I was to receive the 285s. I asked for them to at least rotate the tires correctly so that the 255s are front and 285s on the rear. They claimed that the one 285 rim was not fitting correctly on the rear right, so they left it as-is. Again, I was told that it would be ok until the 285 order arrived.
The 285s arrived on 9/23 and had them installed....no problem...now the rim miraculously 'fit' on the rear.

Next day...9/24...all gears are 'out' on my X5 while driving with a bad whine in each gear. I pull over, not even the "park" gear would work, so I had to use the parking brake. I've been concerned and worried that it could be the transmission. HOwever, today the shop called and said the trans is just fine, that the problem is in the transfer case and will have to open it up.

As I read previous posts concerning the transfer case, AWD, and having the tires correct....Could the 200 miles I've placed on the car since picking it up at the dealer with the tires incorrectly installed have some bearing on damaging the transfer case??? I would think the 'professionals' at Brakes Plus and Discount Tire would know to check specs on a vehicle they are working on. I'm not the 'vehicle professional' here....and was told twice that it would be ok (the tires not properly installed)

Or is it just random for the transfer case to fall ill? I've got 73k on the X5.

I will update when I get the complete diagnosis.... Thank you in advance for any info or suggestions!!