Since it is intermittent at best at this point and there is no coolant leak, I would not expect it to be a water pump issue, or low coolant. Suggest a check if belt is in good shape. A bad thermostat usually sticks open which would not cause overheating. I test a mechanical fan after the engine has been at operating temp to see how easily it turns by hand. Should be some resistance, won't spin even a full turn and feels smooth. First let the engine run at operating temp for 10 minutes or so without the air on and again with it on. The higher the outside temp the better. If it overheats when not running carefully remove the cap and confirm temp. I think more troubleshooting is in order before buying parts and easy to this before your trip.

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