If you were recording it, I suspect if you got a mechanical license for the tune, you could do it and not worry about legal problems. My former singing partner did a political song to the tune of Patrick Sky's Separation Blues. She did the proper licensing through Harry Fox agency and paid the fee. She did find a phone number for Patrick Sky and talked to him as well, though I'm not sure that's legally necessary. I know back in the day jazz musicians would play the same tunes and change a note and a title and there wasn't much copyright fuss.