That meter is suspiciously familiar, be careful where you sing that!
In all seriousness, I wonder how far removed from the original you'd have to go to be recognizable as "that birthday song" and still avoid a copyright claim? Obviously your words are different, but the subject is the same, the meter is the same, and if you stayed in the same key, all you've really done is some word play.
Maybe draw out your "day", "X's", "day", and "all" for each line into a half or dotted half note, and reverse last two notes of each line so they ascend instead of descend. That way most of the meter is preserved and it still feels familiar, but hopefully no longer infringes.
This reminds me of the "Wheel... of Money" parody by WKUK
Every time they cut to commercial or came back from it, the audience had to get a new reminder or instruction from the host on how the way they were chanting "Wheel... Of... Money" was just too similar to another gameshow's name, and the legal team was advising that they would have to chant the name a different way.
So it became "Wheel... [awkward even longer pause] Of Money! [second phrase in all one breath without pausing]"
The contestants also had a problem remembering to let go of the wheel when they spun it, and by the end of the skit all of them had both their arms torn off by the spinning wheel.
I think for the Futurama Birthday song, they probably got away with it, given it was also a parody it has probably hit multiple bullet points on the checklist of whether it should qualify as meeting standards for fair use.
* That means that the owners of Futurama should own the copyright to these lyrics, so they're only free for their use. They could have a copyright claim if you go around singing their lyrics.
* The melody of "Happy Birthday" come from an older song called "Good Morning to All" which is why the tune itself (independent of the lyrics) is probably okay to use. [1]
I have known of other businesses that made up their own lyrics with the same tune for this reason.
In all seriousness, I wonder how far removed from the original you'd have to go to be recognizable as "that birthday song" and still avoid a copyright claim? Obviously your words are different, but the subject is the same, the meter is the same, and if you stayed in the same key, all you've really done is some word play.
Maybe draw out your "day", "X's", "day", and "all" for each line into a half or dotted half note, and reverse last two notes of each line so they ascend instead of descend. That way most of the meter is preserved and it still feels familiar, but hopefully no longer infringes.