Using a microcontroller+BLE (Bluetooth Low Energy) +Servo with one battery source

I am currently working on a project where I was looking for a microcontroller that is capable of BLE (Bluetooth Low Energy) which determines a servo's position.
Using an Arduino the project works well on a breadboard.

To have a better design of the project and enhance the energy efficiency (I need to use batteries and no breadboard)
the Adafruit Trinket was recommended as once
covered within this forum (For a 'customized' arduino consuming less energy making use of Attiny85 vs PCB Fabrication?). As I read I need to use the Adafruit Trinket Pro since the normal trinket with an attiny85 microcontroller will not enable me to attach a BLE module.

Researching on Trinket Pro 5V I found out that the onboard 5V regulator can provide only 150mA output and not 800mA unlike an Arduino Uno.
Therefore I cannot attach the servo(250mAh/4,8V/6V) on the Trinket.

Now I was planning to make use of ONE battery box (6V) and attach it on the Pro Trinket, BLE Module and Servo.
I don't see any other good solution given the requirement that I need to use only one 6V battery box.
I am aware of the risk that the if the Servo uses too much current the device will not work anymore due to the lack of energy. However I will make use of Sleep and interrupt functions to increase the lifetime of the device.

Given the fact that I require a collaboration of a microcontroller, a BLE module and a servo motor, is there a way to use them together in a beneficial/recommendable way?? As of now I consider to attach all of them on an Adafruit Trinket Pro.
However I guess I am thinking too naively; especially considering the battery requirement.

There was a similar question but I cannout extract a useful answer for me from the answers provided there (Using adafruit Flora or Adafruit Trinket + MPU-6050 3-Axis Gyroscope + nRF24L01+ together)