Yes, there are wizarding fairy tales not written by Beedle the Bard.
The author of a reworked fairy tale collection for children, Beatrix Bloxam, took several old stories, including some of Beedle’s, and reworked them so they would be more suitable for children.
“Mrs. Bloxam took a variety of old stories, including several of Beedle’s, and rewrote them according to her ideals, which she expressed as “filling the pure minds of our little angels with healthy, happy thoughts, keeping their sweet slumber free of wicked dreams, and protecting the precious flower of their innocence.”
- The Tales of Beedle the Bard
Bloxam reworked a variety of old stories, and though she did include some of Beedle’s in it, her collection of old stories wasn’t solely retellings of Beedle’s tales. This shows that there had to be other old stories that weren’t written by Beedle.