Wifi versus Bluetooth ?

My gopro battery is discharing very fast and I was told to turn off wifi which uses a lot of energy.
However, when I turn off wifi, I still see on my iphone that it is still connected to GoPro with bluetooth.
If I turn off wifi to save battery, it would mean that I have no way to use the Falcon stalilisator and therefore turning off wifi would not be a good solution.
But bluethooth seems to be used solely to connect the GoPro to the Iphone using Quick, and not to connect the Falcon to the GoPro.
Can someone tell me what to do to save battery whilst connecting the gopro to the Falcon ?