I didn’t mean to emphasize the Lattice, but the fact that the curve appears to ‘sink’ into the surface. That’s why I added the pictures…
You have snapping to surface enabled, but what are your settings? For instance compare:

to this:

In the first picture the curve disappears in the (curved surface of the) sphere, because it snaps only one point of the curve to the surface of the sphere.
In the second picture it snaps each ‘curve vertex’ to the surface.