<rant on>
There must be something I don't understand about Godot's multiresolution stuff. Godot seems tuned for making "pixel perfect" games, but as soon as you don't care about that, life gets hard. For example, all the sprites are rendered at 4x the required resolution for 1080p, so they should be able to look smooth and pretty on any modern display - but I couldn't figure out how to tell Godot to increase the resolution rather than scaling the viewport at a fixed resolution - and it looks like I shipped with the wrong setting on that as well.
<rant off>
Sorry about that, and if you know what I need to change to get multiple resolutions to work, I'd love to hear it.