If it was me, I wouldn't start with the tires. They would potentially cause a problem with the transfer case, not the transmission. And the codes would be related to the transfer case.

Neither would I focus on the DME or TCM.

You can conclude from the results when you are manual shifting that the clutches/etc can manage a shift when the transmission doesn't have to decide when to shift.

I would be looking to the transmission valve body, transmission wiring harness, speed sensors, actuators, solenoids, and other monitoring/control systems internal to the transmission that manage shifting.

Recall that failsafe mode is just the transmission protecting itself because it doesn't know what to do, based on lack of inputs from sensors, or inputs out of the expected range, or whatever.

You need a test plan that guides you to the likely problem components from the codes you are getting.

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2007 X3 3.0si, 6 MT, Premium, White

Retired:
2008 535i, 6 MT, M Sport, Premium, Space Grey
2003 X5 3.0 Steptronic, Premium, Titanium Silver
2002 325xi 5 MT, Steel Grey
2004 Z4 3.0 Premium, Sport, SMG, Maldives Blue