I never realized that was a regional expression. I thought it was just another one of my dad’s nonsensical expressions, like, “well rain on their bald heads,” meaning, “to hell with them all.” He used it in the context of planning to take some unpopular action, “I don’t care if it hare-lips the mayor.” It sounds as if whomever is getting hare-lipped is optional, as long as it rhymes?