Project quad onto ellipse in 3D
I assume that your texture is deemed to belong to the plane of the trajectory.

Looking at the unprojected ellipse, you can map the circle to it by an anisotropic scaling. If necessary, rotate the texture first if there is any orientation requirement.

To achieve this first mapping, you will compute the four corners of a rectangle, presumably related to the frame formed by the ellipse axis.

Then transform those vertices to the global scene coordinates, the same way to transform the ellipse.