I think you can run it at 225 indefinitely. The transmission fluid will be fine.

I think the alarm point is around 260, IIRC.

When you look for references for standard temperatures, you want to look at modern ZF automatics and not older lower tech transmissions with older fluid specs.

And one of the links you included is selling transmission coolers.

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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