I truly don't know, but I imagine that if you were wearing birthday hats, consuming cake or some other celebratory baked good, and clearly not worshiping an Egyptian river god, there may be a problem or two convincing someone you weren't singing the copyrighted words.

No, because it is obviously a derived work. It's extremely unlikely someone who never heard the tune "Happy Birthday" would come up with lyrics for the (non-copyrighted) melody that sound exactly like the copyrighted lyrics by sheer coincidence.

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

Given that Robin Thicke got sued just for writing a song that kind of has a little bit of the same feel as a Marvin Gaye song, I doubt this works.