Curved surfaces are more complex as they aren't just a 2D plane. Which means that mapping to extents doesn't work as your curved pixels are no longer 'square' to the camera you're mapping from.
If you're using LED panels, or a correctly faceted curved mesh, then you can use mapping regions to get the correct content sections, and then use the 'swing to selection' function with each column selected to get a perpendicular view to a section and then map to extents. You will need to create a new region for each facet of the curve, which can get a bit tedious. OR, you can...
Create a custom mesh, in something like blender, and apply a correct UV map so that content is mapped correctly to the mesh. You can then import that to capture (as an .obj) and simply replace the texture source being used, and it should now be corectly applied to the mesh. Do not touch the material mapping as that will break the UV map and revert to Capture's normal 2D projection method.