Is the mentioned report authentic, and does it disparage the Companion Mu'awiyah, may Allah be pleased with him?

The story mentioned in the question has no basis in the books of Sirah (Prophet's biography), biographies, history, or the well-known Sunnah. Rather, it was mentioned by some literary figures such as Abu Sa'd al-Abi, Ibn Hamdun al-Baghdadi, and al-Abshihi. It is a story that does not befit these two venerable Companions, may Allah be pleased with them. It is not permissible to take from books of literature and rhetoric in such matters, nor should they be relied upon, due to the great harm they contain regarding the early generations and the Companions of the Prophet, peace and blessings be upon him.