How authentic is the anthem "Tala' al-Badru 'Alayna" attributed to the Prophet, peace and blessings be upon him, and is it permissible to chant it?

The chanting of the verses "Tala'a al-Badru 'alayna" (The full moon rose over us) has been narrated in books of Sirah (Prophet's biography) and some books of Athar (traditions), but its chain of narration is weak and disconnected (mu'dal). Ibn al-Qayyim stated that there is a misconception in it because "Thaniyyat al-Wada'" (the Farewell Thaniyyat) is located on the side of Al-Sham (Syria), and would not be seen by someone arriving from Makkah to Madinah. Despite the weakness of its chain, there is no harm in mentioning these verses, narrating them, and chanting them, because their meanings are sound and good, and their fame is widespread. Furthermore, there is nothing in them attributed to the Prophet, peace be upon him; rather, they are from the words of the Companions. Scholars have shown leniency regarding narrations in Sirah, stories, and sayings of the Companions and the Tabi'in (successors) when no doctrine or Sharia ruling is built upon them.