My next idea was to make a pattern swatch, hit expand, and tweak it with envelope meshing (Object>Envelope Distort>Make with Mesh) but the result was too warbly.

At this point, I was pretty frustrated. I was convinced that Illustrator just didn’t have what I wanted it to have. Dave was convinced that there was a way to do it, so more googling happened. The thing that worked was something I never would have thought of. Select a group of objects, and hit Effect>3D>Extrude and Bevel. Turn off all of the shading and bevelling, and finally there was something you could set a perspective to.

And this was the result that we were looking for