Bending an object over a surface

Hummm I need to think how to use to use it on this case since the target and projection both need to be quadrangular and I have a round surface in both cases. That is, a circular mesh and the ‘projected’ circular mesh.

EDIT:
After much pondering I don’t think it will work.
The thing is that there is no plugin that I know off, for projecting one plan surface onto any kind of curved surface, including a sphere or a dome with an open angle above 180º.
Easiest way is indeed with a texture and Fredo’s ThruPaint but that won’t give 3 dimensionality of course.
It would be nice that someone could develop such extension. Programmatically I don’t think is such a big deal.