I waited oh-so-patiently for the sugru to set. The full 24 hours, and then an extra night's sleep to be on the safe side and to share the trial run with Steve (only fair since it was him who sourced the doorbell itself and created the mp3 for it to play).
The sugru blob I'd made - thick enough to accommodate the smiley face drawn onto it - was, when set, too thick to be flexible enough to push the doorbell-button through it with a single finger. It was also too large in diameter to press the button with it, as the whole red circle couldn't go into the casing.
I haven't explained that well, but the upshot was that the only way to press the doorbell was to hold the unit in your hand and squeeze as hard as possible. Not really practical.
In retrospect, the sugru blob needed to be smaller than the original button, or thin enough to be bendy, or both. At least, unlike with the daffodils I ruined, I know what I did wrong.
Thankfully, the folks at sugru are aware that their products may be used by the inept and hard-of-thinking and give tips on their website for how to remove it. A few minutes of running my fingernail around and around the red button loosened it enough for me to be able to peel it away.
As a happy side effect, the previously white button has taken on some of the red hue from the sugru, making it visible - which was the original aim.
I've managed to correctly spell my name, so that's something.