Using a microcontroller+BLE (Bluetooth Low Energy) +Servo with one battery source
The Atmega328P can actually use a supply of up to 5.5V, so using a silicon rectifier to reduce the 6V and a bulk capacitor (after the diode) to help with brownouts will work, and the servo will be supplied directly off the 6V battery. If your BLE module needs 3.3V then you will have to use a separate regulator for that.