OK! I think I have the answer to my own question! With the quad selected, in the quad inspector on the bottom left, select "Mesh Warping" and then "Show Input Mesh"--this will let you determine what part of the INPUT you want to use. The key is to do the "mapping" part of it IN THE INPUT, while watching the effect on the output. So mapping with the input doesn't affect anything on the output except what portion of the input you're seeing, whereas "normal" mapping on the output will affect scale and perspective.
Giving credit where it is due: