Patch on a sphere of varying size

Imagine a patch glued to a sphere. How would I manage to make the patch keep its center position and surface area as the sphere is scaled up or down? Normally, only the curvature of the patch should change, as it is « glued » to the sphere. Assume the patch is described as a set of ( latitude, longitude ) coordinates.

One possible solution would consist of converting the geographical coordinates of the patch into gnomonic coordinates (patch viewed perpendicularly directly from above), thereby making a 2D texture, which is then scaled up or down as the sphere changes its size. But I am unsure whether this is the right approach and how close of the desired effect this would be.

I am a newbie so perhaps Unity can do this simply with the right set options when applying a texture. In this case which input map projection should be used for the texture? Or maybe I should use a 3D surface and « nail » it somehow to the sphere.

Thank you!