Hey Jarrett and welcome to the forums,
as far as I know the RGB output of asus motherboards is 12 V with a max power of 2A. It supports only 5050 RGB LED strips, so I think your cooler will also use a built in 5050 LED strip.
As the BlinkStick Pro itself controls WS2812 LEDs (up to 5V) it will not be compatible to your cooler.
I´m not familiar to 5050 stuff and the asus RGB functions but maybe it works with a LED adapter board:
You could give it a try but I´m not sure about if your cooler will work with that.