Had to go ahead and make a new project, as packaging the main project was taking too long. Anyway, the “I” behavior was identical between the play in editor and the packaged versions. Curiously, the packaged version responded to the Escape key exactly as intended, removing the widget instantly.
To package, I just went to File > Package Project > Windows > Windows (64 bit), set the destination to a folder on my desktop, and let it do its thing.