After counting down 5 minutes, the badges go into a deep sleep mode.
Previously that was using around 225μA, which was a little bit too much for something that was supposed to be in a deep sleep.
After some additional tweaking of the existing code I managed to get the ATTiny using just over 6μA. That's 37 times less power being used.
Using a standard CR2032 battery 40 times a day, every day, puts the estimated battery life at: 80 days!
Obviously if used more frequently it wouldn't last as long.