I actually just got it to work!
I'm pretty sure the Gamecap came with a faulty USB cord. I noticed that the light on the device itself sometimes changed from purple to red when jiggled around in the Gamecap's port.
I still think this is strange though, because as you said, the red no signal means that the software recognized that the Gamecap was connected to my computer.
Also, the order in which I connect wires seemed to effect whether the device works. When I connect the Gamecap to my computer while both HDMI cords are inserted, it doesn't work and shows a red no signal, even with a new cord. But if I connect the Gamecap to my computer first with the new cord, then my TV to the Gamecap second, and third connect my Switch to the Gamecap, it works fine.
Thanks a lot for your help in setting this up