The transmission operating temp is reached in very few miles or idle time. I don't think idle or A/C is part of the problem. I doubt that it has anything to do with trans temp. But, as suggested, use a infrared temp tool to read trans fluid temp cold and let it idle for 10 minutes and read it again. Normal operating temperature ranges from 176-212 F. Start by reading temp, easiest and cheapest thing to do. It is not unusual that the 2-1 rough shift does not happen in sport mode.

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X5 4.6 2002 Black Sap, Black interior. 2013 X5M Melbourne Red, Bamboo interior
Dallas