When you use Image>Transform>Flip, you flip the image and so flip everything in it: layers, channels, paths, the selection mask..... What you want is the Flip tool in Path mode.
And maybe you want to have a look at my path-mirror script that improves on this, because it can splice together the two halves (the original and the flipped copy) and create properly closed strokes.
And maybe you want to have a look at my path-mirror script that improves on this, because it can splice together the two halves (the original and the flipped copy) and create properly closed strokes.