I've been testing run times with the rechargeable battery. I am running the tinyScreen+ with the BLE board and the screen set to a brightness of 10. I was able to get 18 hrs on a battery before doing any power management/optimization.
You would not be running any comms (wifi, bt etc..) so your mileage should be better than that.
If you keep the main loop fairly empty, and redraw as little as possible (don't redraw on each loop, either check time, or even be super cheesy and do a count down, redraw) you should be able to stretch the battery out.