Uthman was absent from Badr only because he was married to the daughter of the Messenger of Allah ﷺ and she was ill. The Prophet ﷺ said to him, "You will have the reward and share of one who was present at Badr."
إنما تغيب عثمان عن بدر، فإنه كانت تحته بنت رسول الله ﷺ وكانت مريضة، فقال له النبي ﷺ: إن لك أجر رجل ممن شهد بدر وسهمه.
Bukhari narrated it in the chapter on the obligation of the khumus (3130), from Musa, who said, Abu Awana narrated to us, Uthman bin Mawhab narrated to us, from Ibn Umar, and he related it.Abu Dawud also narrated it (2726) from Mahbub bin Musa, who said, Abu Ishaq Fazari informed us, from Kulayb bin Waail, from Hani bin Qays, from Habib bin Abu Mulaykah, from Ibn Umar, who said, The Messenger of Allah ﷺ stood up, meaning on the day of Badr, and said, "Uthman has gone out on an errand for Allah and an errand for His Messenger, and I am pledging allegiance on his behalf." So the Messenger of Allah ﷺ allotted him a share, and he did not allot a share to anyone else who was absent besides him.Its chain of transmission has no problem with it. Hani bin Qays was narrated from by a number of people, and Ibn Hibban mentioned him in his book of trustworthy narrators.