I tested the self latching circuit using a scrapped PCB from an old project of mine. This PCB was already having the proper pads for the TSSOP microcontroller. 

I powered the circuit exclusively from the coin battery in order to be able to check if that can provide enough power. I also did a leakage current test with a bench Keysight 34461 multimeter. The measured stand-by current was less than 1nA. The current absorbed by the circuit in normal functionality, varied between 5mA and 8mA, depending on the image on the screen. According to the datasheet, the capacity of the battery is around 70mAh and this means that it could power the circuit for an average of around 10h. If the scrolling message lasts around 12s, this can result in being able to display the message roughly 3000 times before the battery drains out.

I am really happy with this test. This means that I can now order the PCB from the supplier.