Path start point of Sketch on Path

The start point (at 0% arc length) of a Sketch on Path is usually the end of the path curve.  However, I'd like it start at an intersection point where another curve crosses the path.  The only way I know to do this (in principe) is to break the path into two curves at the intersection point.  However, the path comes from a mirror operation, so that gets tricky with regard to parametric changes and updates. Are there any other ways to do this?  Thanks for any suggestions.