To what extent is the hadith (narration) "The closest of you to me in rank on the Day of Resurrection are those who send the most blessings upon me" authentic, and is its chain of transmission sound?

In summary: The Hadith, "The most frequent among you in sending blessings upon me will be the closest to me in rank," was narrated by Al-Bayhaqi, and it contains two defects: Makḥūl Al-Shāmī did not hear from Abū Umāmah, and Ibrāhīm ibn Al-Ḥajjāj may be a distortion of Ibrāhīm ibn Ḥayyān, who is weak. As for the Hadith, "Indeed, the closest of you to me on the Day of Judgment in every place will be the most frequent among you in sending blessings upon me," it was narrated by Al-Bayhaqi and includes "Ḥakāmah bint ʿUthmān ibn Dīnār," whose narrations are abandoned. And the Hadith, "The people most deserving of me on the Day of Judgment are those who send the most blessings upon me," was narrated by Al-Tirmidhī, and it contains Mūsā ibn Yaʿqūb, who is truthful but has a poor memory, and his شيخ (shaykh) ʿAbdullāh ibn Kaysān is unknown. The result is that this text has not been narrated with a sound chain of transmission, although some scholars tend to strengthen the Hadith through its various طرق (chains of narration) and classify it as ḥasan li-ghayrihi (good due to external corroboration).