There are only two things left to implement — LCD display functions and basic control functionality: sound switch, backlight, buttons for switching display mode and resetting the dose. I don't see much point in explaining how it works, let's get to the results instead. The total current consumption when counter is idling is 44 µA, which is low enough to avoid the battery capacitance derating. Given that the speaker is disabled, and battery capacitance is 150 mAh (the device stops working at 1.8V, so we can have individual cells discharged to 0.9V, utilizing their full capacity), this gives us a total working time of 3409 hours or 142 days or 4.7 months. This is a best-case number, so I've rounded it up to 4 months. I've left the device turned-on and will post the actual number of days it managed to last. At the moment of writing this, it has been working for 12 days non-stop without any problem and the battery loses around 8mV per day.
I'm happy with the code performance too. The whole update routine (radiation rate, dose calculations, time update and LCD print function) takes only 16 ms or 524 clock cycles. We save some power because the CPU spends so little time in the active mode. A friend of mine decided to rewrite the firmware in full assembly, optimizing it even further. I will post an update if he will actually do it.
Here's some photos and a video of a finished device. It was designed with an enclosure in mind but apparently, I won't be able to make it any time soon.
Source code and schematics are available on my Github.