Muḥammad ibn Manṣūr is reliable, and Manbūdh was declared trustworthy by Ibn Maʿīn. ʿAbd al-Razzāq, Ibn Abī Shaybah, and al-Ḍiyāʾ also transmitted this report from her with virtually the same wording in al-Mukhtārah.
The ḥadīth has further corroborating chains.
I add: al-Taqrīb describes Manbūdh’s mother as “acceptable,” and it gives the same verdict—“acceptable”—for her son. With such supporting reports, the chain is therefore ḥasan (sound) as corroborated.
Al-Shawkānī then said:
“This serves as proof for those who permit a menstruating woman to enter the mosque when there is a need, and it supports linking the prepositional phrase in the first ḥadīth to his words ‘Hand it to me,’ for her entering the mosque to place the khumrah (small prayer-rug) inside is no different from her entering it to take it out.”
Among the corroborating narrations to which al-Shawkānī referred is the report from Ibn ʿUmar:
The Messenger of Allah ﷺ said to ʿĀʾishah, “Hand me the khumrah from the mosque.” She replied, “I am menstruating.” He said, “Your menstruation is not in your hand.”