I have a wicker basket which is made up of curves. The object has a couple of array modifiers, a simple deform and a cast modifier to achieve its final shape.
The object on the right had to have its bevel depth reset to 0 before the modifiers were applied; it was then converted to mesh and then back to a curve again before re-adding the bevel depth. With this method it successfully bevels the curve without any distortion to its profile. This is the desired effect I'm not able to reproduce using a non-destructive workflow.