All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad, sallallaahu 'alayhi wa sallam, is His slave and Messenger.
The scholars differ over whether this hadith is authentic or weak. Al-Hakim said in al-Mustadrak: “This hadith has a sound chain of narration (sahih al-isnad), though the two Shaykhs [al-Bukhari and Muslim] did not record it,” and al-Dhahabi agreed with him in al-Talkhis.
Ibn Hajar
said in Nata'ij al-Afkar, quoting from al-Kamil by al-A'zami: “This hadith is hasan (good)... and its narrators are the narrators of the Sahih, except 'Abdullah ibn al-Walid, for he is an Egyptian about whom there is disagreement.” [End of quote]
Al-Hilali
said in 'Ujalat al-Raghib: “Rather, its chain is weak, as this 'Abdullah ibn al-Walid is Ibn Qays al-Tujibi, and he is weak...” [End of quote]
Ibn al-Amir al-San'ani also graded it weak in al-Tahbir, as did al-Albani in Da'if Mawarid al-Zam'an.
So the hadith is disputed as to its authenticity, as you have seen.
As for making this supplication, whether in prostration (sujud), after the tashahhud in the prayer, or outside the prayer, there is no harm in it. It is permissible to supplicate with reported prophetic supplications and others, all the more so when its origin lies in the supplications of the Noble Quran, whether from a single verse or several verses.
Allah Knows best.