could I just add a microcontroller or timer to this transmitter circuit to turn it on and off so that it lasts longer?
Using a microcontroller to make a low power timer is a standard thing to do.

Note that there are several techniques to reduce the microcontroller power, all of which can be found with the use of your favourite search engine. These include 

a) choose a microcontroller which has app notes describing how to run at very low power
b) use a low voltage
c) use a slow clock (kHz not MHz)
d) leave the microcontroller in sleep for most of the time with only the RTC running