I do not have servo for 2AA batteries. That is currently the power source only for the ATMEGA328P-PU with the NRF24L01+ module. I am going to power the servo with separate power source which is most probably going to be 4 to 6 AA batteries. The server I will drive is here.
It is right, that I can power up the BTS or BTT with this power source too, and only switch the load on and off with the arduino powered up with 2 AA batteries. Not sure if the BTS or BTT can use the logic of 1,8 V to switch on and off.
By the way I have one more idea. Probably not the best solution from electronics point of view but I will try to describe here. I have some power bank at home which is designed to provide 5V with 10000mAh of capacity. If I would connect it with the servo motor then it should be able to run for about 100 days in lets say stand by mode. I mean it could operate for 90 days if I would not run the servo motor (just powering it up as it is taking 4mA when not running if I am correct). The problem but in this case also an advantage is that the power bank switch off automatically when the powered device is not taking enough current on the supplied USB port. It means when the servo need to take the power from the power bank, the power bank should be first wake up by some current, and then just turn the servo as needed. After that the power bank will again turn off when no current on the USB port. I already tested with my power bank and looks ok. So no I just have to find a way how to wake up the power bank to provide power source for lets say 2 seconds to turn the servo. I was thinking to just connect an optocoupler with some resistor to simulate an power requirement. This would be driven by the ATMEG328P chip. I was also thinking to use this power source from power bank also for the ATMEG328P chip. My idea was to connect there an supercap (I have one for 2,5V with capacity of 10F) which would just take the power from power bank (via some diode to reduce the voltage from 5 to 2,5 V) and then will supply with this power the ATMEG chip which will be mostly in sleep state. I am sure this is not the best idea ever 
but I will probably try as I have all the components at home.
Anyway I would prefer some more easier solution in form of a simple board with the BTS or BTT and ATMEG+NRF. But I would need some help probably to design it. Maybe it could be useful for others in the world of Arduino too as a solution for IoT with servo needed projects.
What do you think MorganS?