Grading: (2) Its chain is very weak (da'if jiddan) because of "Ibrahim bin 'Uthman" - who is al-'Absi by clientage, Abu Shaybah al-Kufi - and he is "matruk al-hadith" (abandoned in hadith).
Corroborating chains and supporting reports: However, he is not alone in narrating it, for this report will come shortly with the author under no. (4899) via Muhammad bin Yunus al-Kudaymi, who narrates from Abu Zayd Sa'id bin Aws, from Shu'bah, from al-Hakam.
Technical note / Defect ('illah): But (the narrator) "al-Kudaymi" is also very weak, so this corroboration carries no weight. It is also narrated from Ibn 'Abbas through another chain, but that too is very weak, so it is nothing to rejoice over (i.e. it cannot be used as evidence).

Important point: Although this is the state of its chains from Ibn 'Abbas (as explained), this report is "well known and famous (mashhur ma'ruf)". Seeking a chain for such well-known matters is excessive (takalluf).
Details of the narration: As for the mention of the Prophet's (peace and blessings be upon him) daughters, it is "agreed upon", with no two people differing over it. As for his (peace and blessings be upon him) sons from Sayyidah Khadijah (may Allah be pleased with her), something similar is narrated from Mus'ab bin 'Abd Allah al-Zubayri and his nephew al-Zubayr bin Bakkar, and it is also narrated from al-Zuhri.
Reference / Source: Ibn 'Abd al-Barr said in "al-Isti'ab" (p. 889): "This is the view of most of the genealogists." And Ibn 'Abd al-Barr quoted the genealogist 'Ali bin 'Abd al-'Aziz al-Jurjani as saying: "This is what is correct, and anything else is confusion (takhlit)."

Reference / Source: Ibn Nasir al-Din al-Dimashqi quoted it in "Jami' al-Athar" [3/ 468 and 469] from Abu al-Hasan 'Ali bin al-Junayd al-Razi, Abu Bakr al-Barqi and Abu al-Faraj Ibn al-Jawzi.

Reference / Source: And this narration is found in the author's book "Fada'il Fatimah al-Zahra'" (76).

Reference / Source: It was recorded by al-Bayhaqi in "Dala'il al-Nubuwwah" [2/ 70], and via him by Ibn 'Asakir in "Tarikh Dimashq" [3/ 140], from Abu 'Abd Allah al-Hakim, with this same chain.

Reference / Source: It is found in Yunus bin Bukayr's additions (337) to Ibn Ishaq's "al-Sirah al-Nabawiyyah", and via him it was recorded by al-Dulabi in "al-Dhurriyyah al-Tahirah" (47) and Ibn 'Asakir [3/ 124 and 12/ 124].

Reference / Source: Al-Tabarani recorded it in "al-Mu'jam al-Kabir" (12115) and "al-Awsat" (1463) via 'Abd al-Salam bin Rashid Abu al-Hasan al-Raffa', from Ibrahim bin 'Uthman, likewise.

Reference / Source: Ibn Sa'd narrated it in "al-Tabaqat" [1/ 110 and 3/ 6], and via him Ibn 'Asakir [3/ 125] and Ibn al-Jawzi in "al-Muntazam" [2/ 316], from Hisham bin Muhammad bin al-Sa'ib al-Kalbi, from his father, from Abu Salih, from Ibn 'Abbas.
Details of the narration: And in it he said: "In Islam 'Abd Allah was born to him, and he was named al-Tayyib and al-Tahir."
Status of the narrator: But Hisham and his father are both "matruk" (abandoned), and Abu Salih - who is Badham, the freed slave of Umm Hani' - is "da'if" (weak).

Details of the narration: Besides this, it is narrated from Ibn 'Abbas through another "very weak" chain, as Ibn 'Asakir [3/ 128 and 12/ 128] narrated via Muhammad bin Zakariyya al-Ghalabi, from al-'Abbas bin Bakkar al-Dabbi, from Muhammad bin Ziyad and al-Furat bin al-Sa'ib, from Maymun bin Mihran, from Ibn 'Abbas, that: "Khadijah bore 'Abd Allah, Zaynab, Ruqayyah, al-Qasim, al-Tahir, al-Mutahhar, al-Tayyib, al-Mutayyab, Umm Kulthum and Fatimah."
Status of the narrator: Muhammad bin Zakariyya is "matruk" (abandoned) and al-'Abbas is "da'if jiddan" (very weak).

Reference / Source: As for the narration of Mus'ab bin 'Abd Allah al-Zubayri, it was narrated by al-Bayhaqi in "Dala'il al-Nubuwwah" [2/ 70] and Ibn 'Asakir [3/ 130].

Reference / Source: And as for the narration of al-Zubayr bin Bakkar, it was narrated by Ibn 'Asakir [3/ 131].