Your fan controller is controlling the frequency supplied to the whole fixture, both light and fan motor. When you make the fan turn faster or slower, you’re affecting the current supplied to the light as well. I would guess this could potentially damage either your bulbs (particularly if they’re LED) or the Hunter controller supplied with the fan.
If there isn’t another wire already installed in the wall to power the light separately, the only easy option you have is the Hampton Bay Wink-enabled fan controller. This would replace the manufacturer-supplied controller and give you separate control of the fan and lights with a handheld remote and via SmartThings. When installing that, you would remove the GE fan control and either put your dumb switch back in the circuit or wire the fan to always have power. Just note that you’ll need a Zigbee repeater (smart plug) in the same room as the range on the fan controller is very short.