The ST-10 is not compatible with the Breeze. The ST-10 is a radio controller. The controller for the Breeze is a Bluetooth device that sends control signals to your phone, that then uses WiFi to actually control the Breeze.
Some have side loaded the Breeze Cam app on an ST-10 which is Android based, but the sticks are unusable so it is like using an oversized phone for control.