I found the source of the issue and a workaround!
I noticed that the system audio being played through the input was lower in volume, which I thought was weird. It definitely wasn't leaking through my headphones, or it would have sounded tinny. This led me to wonder why the interface was presenting itself as a 4.0 surround device considering that it's only supposed to be stereo. On a whim, I moved the sound output fade slider all the way to "Front", and it stopped feeding through system sounds! I assume the extra 2 channels the MiniFuse presents are some kind of internal loopback?
This isn't a perfect solution, however. It still doesn't present itself as an input device in the Settings application (despite working just fine), and the fade slider doesn't save its position if you reopen Settings (but it continues to work as if it was). Functionally, it's perfect. But the user experience is definitely not good.