I'm trying to create a "cable" (i want to animate the start/end-point of it later) on a surface.

As you can see the shrinkwrap modifier is doing a pretty good job: Shrinkwrapped Curve

But as soon i try to bevel the curve it looks completely destroyed: Shrinkwrapped Curve with Bevel that looks very very bad

Is there a way to maintain the curve without affecting the geometry?