It was a bust. The button cell I selected just can't supply the required current. When I hooked up a pair of AA batteries, it worked well.

Later, when I tried to substitute a 1F capacitor (thinking "tiny *rechargable* regulated LED flashlight) I screwed up and either damaged the tiny SMD button or bridged some pads under it, because now it is stuck "on". Oh, and it doesn't run long from the capacitor anyway.

I still like the idea, but I need a different power source. Oh well.