The first prototype is complete. With the next revision I will aim to minimize the current draw to extend the time. If this were to become economically feasible, I will need to use less super caps, possibly down all the way to just one 15F.

Run Time Extending Changes:

-rather than have my output of the regulator at 2.75V I can increase that to 2.8V (voltage rating of supercaps).

-adding a diode between the output of the regulator and the caps will ensure that no current flows back through the regulator circuit when the supply is removed. The resistor divider network in the regulator would draw a little over 2mA passively. With a diode in place there is no way for the caps to back-feed into the resistors.

-using a lower power timer/osciallator. My 555 timer has a listed current draw of 250uA, while there are some available I've seen that are listed as low as 50uA.

-use LEDs with a lower current draw and 1.8V forward voltage.

A Hackaday user also recommended a buck/boost converter to use to bump the voltage back up once it falls below a usable value. This is something I will look into as well.