It's not that you need to worry about the wrong tire size, but rather if the tires are different heights front and rear. Generally those would lead to a blown transfer case, not transmission.

fwiw, my re-manufactured trans was $4050 from BMW.

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2002 X5 4.6is - Titanium Silver / Black Alcantara - Dinan Stage 2 - Bilstein/Eibach - Brembo BBK ----SOLD---
1993 318i - Brilliantrot / Tan - bone stock daily driver