In which I resolve this issue.
Using the Android NotificationListenerService API, and a little help from Ketan Parmar's example code, I got the basic proof-of-concept running.
Then I had a bit more work to do, because of some typecasting issues.
Should-be-signed APK available here. (Note: This may be out of date.)
Made available under the MIT License. Do whatever you want with this.