With straight sharpening methods such flimsy image elements can be problematic - the selecting
being one. (Only as an aside, the unsharpened version looks pretty natural to me.)

For such fine lines, perhaps first carefully select a small area, then use the Blur/Sharpen brush,
and tidy up with the History Brush. If you look at the pic portion below, the steps are:
1) Set the Magic Wand tool as shown, select bits of adjacent sky, and when ready, INvert the
selection.
2) Apply you sharpening method (The brush shown gives lots of good control), and if you make
some inadvertent small errors, apply the
3) History brush, usually at 100% and Normal to completely reverse it at that local level.

Your pic portion...
shrpening1.jpg