You could just sing about an Egyptian river god instead:
https://www.youtube.com/watch?v=2f2PCWYAZQc
>The melody for Happy Birthday was first penned by two sisters from Kentucky, Mildred J. Hill and Patty Smith Hill. The song was called Good Morning to All, but bore the recognizable melody. The tune was first published in 1893 in the book Song Stories for the Kindergarten. The melody has since passed into the public domain, and is safe to hum in public without permission.
You could, however, use entirely different lyrics. But I'm pretty sure it could still be assumed to be a derived work as long as it's a birthday song.
Disclaimer: IANALATINLA, etc