As others have already said, rebuilding curve is the way to go imo. I've tackled the same problem you are mentioning, and solved it with rebuild curve.
Is supporting edges necessary or would tweaking the vertex normals of the flat part of the blade to all face forward from the surface also solve the issue?
Try using the actual clear mask button under the masking section in tools, rather than crtl+dragging in the viewport. That's solved it for me in the past.