Ok, so all is well cept the damn light comes on. MOST of the time, that is the coolant level sensor. It can be a flaky fan controller as well.
In fact, the ECU controls that light, if it sees any of this, it will turn the light on:

Indication of low coolant level, either cause it's low or the sensor is bad or the connector to the sensor is bad or the wrong sensor was used.
High engine temperature (the gauge will show it too).
A communications error between ECU and FCM.
Indication from the FCM that fan motor current is out of specs (If the fan is not told to run, then this won't be a thing.)
A flaky FCM.
A flaky ECU. (Not likely, never seen it before.)

SOME of these indications will cause the ECU to inhibit A/C engagement. The fact that you have no such failure is interesting. I do not know if the STAR tool would tell you what the ECU is seeing that makes it want to turn the light on.