To answer your question, many all wheel drive vehicles are sensitive to the diameters of the tires, that is they do not tolerate very much of a difference in the diameters in the tires for very long, and by very long, I mean as little as 50 miles.
Neither you or the mechanic that certified the car can tell if the previous owner had run tires with different diameters for very long, i.e. using an odd size tire or replacing just one or a pair of tires instead of all four.
If the previous owner had a blow out when the tires were about half worn and just replaced one of them, it will damage the transfer case on a lot of AWD vehicles. If the previous owner did not do tire rotations as recommended and the difference in tread depth became 2/32" or greater between the most worn and least worn tire, damage would be done. If the previous owner replaced only two tires due to wear, damage would be done.
Check all four of your tires for tread depth. If they are within 2/32" of each other now, you should not have any trouble with your new transmission. If they are greater than this, then you need to either have the best tires shaved to match the tread depth or the most worn one or buy four new tires.
Be sure to keep up with your tire rotations and alway look for signs of misalignment. I do not recommend getting an alignment (4 wheel on this car) unless there is an issue with tire wear or drivability, but at the first sign of either one of those issues, get it aligned immediately. Also keep an eye on the tire pressure, a severely low tire can also cause a strain on the transfer case.