I'll take your word for it, Monique. The dictionary I have does say it's "allégre," so that's probably not what Clooney is singing.    It does, however, give the word "gré" as a noun meaning "liking." So would "Danser à le gré" mean "to dance to my liking" (or, more idiomatically, "to dance for my pleasure" or "to dance any way I like?"

It would be great to find the sheet music and/or a recording by someone else who uses Clooney's French lyrics to the chorus.