Observation shows that the change in the click rate for my Geiger counter has, for now at least, stopped changing (so much) over time. Here's the data from a recent run of well over a day:

As you can see, the average is no longer falling as it was. In fact it's behaving within statistical limits. If it had behaved this way from the beginning, I would not have become suspicious it was changing.

Quite why this should be, I'm not at all sure. Maybe the brand new Geiger counter just needed to run for a while, and now all's well. Maybe it's just not very stable? Maybe its sensitive to "the phase of the moon"? I haven't found anything yet that would indicate it's fundamentally unstable or subject to malign influences, but then I wouldn't: its been stable for a while now.

Such as it is, that's the good news: results so far don't rule out getting this to work. But there's bad news, too. A little research shows that the sensitivity of Geiger counters of the sort I'm using tends change over time. None of the sources I've found say why the sensitivity changes, but people talk about needing to calibrate "survey Geiger counters" against a standard every few months. That doesn't bode well for what's supposed to be a source of truth. I'm beginning to worry I need a different sort of detector.

Research, both into stability and into alternative detectors continues.