I have just received an answer from the seller. According to him, he obtained the badge directly from the pilot himself who also told him the story I posted yesterday. As one had to expect, there is unfortunately no written proof of that.
However, the seller gave me the name and rank of the pilot. I made some research and found that his 'Soldbuch' was sold by a famous German auction house less than a month ago.
I have never seen a Luftwaffe pilot badge in a similar shape for less money. The damage and repair are almost not visible on the obverse. The only thing you see is the crack in the upper part of the swastika, just below the eagle's claw (see 1st and 3rd images in the opening post). If you know someone who is selling such badges for 200 dollars, let me know, I would be interested.