The Prophet (peace and blessings be upon him) did not command Abū Bakrah to repeat the rakʿah for which he had hurried. Had he actually failed to catch it, the Prophet would unquestionably have told him to make it up, for the Messenger of Allah (peace and blessings be upon him) never deferred clarification when it was needed; his duty was to convey the message, and the one who conveys speaks whenever explanation is required. Since the Messenger (peace and blessings be upon him) did not say to him, “You missed the rakʿah,” we know that he did in fact catch it, and in that situation the obligation to recite al-Fātiḥah is waived.
There is supporting legal reasoning as well: al-Fātiḥah is obligatory only while one is standing, and here the standing has been dropped in order to follow the imam in bowing. When the standing falls away, the obligatory remembrance tied to it falls away as well.
Both the textual evidence and the underlying rationale therefore show that whoever arrives while the imam is in rukūʿ should pronounce the opening takbīr (takbīrat al-iḥrām) while standing, recite nothing, and then bow. Should he add a second takbīr for the bowing itself, that is preferable; if he does not, there is no blame, for the first takbīr suffices.
It is mandatory for the worshipper to recite al-Fātiḥah while standing. As for the practice of some—when the imam rises for the second rakʿah, they remain seated and begin reciting al-Fātiḥah, rising only after the imam has reached the middle of it, even though they are fully able to stand—we say to such a person: your recitation of al-Fātiḥah is invalid. Al-Fātiḥah must be recited in the standing posture, and you were capable of standing yet recited part of it while sitting; accordingly, that recitation does not count.
Anything recited beyond al-Fātiḥah is Sunnah in the first and second rakʿahs. In the third rakʿah of Maghrib and the third and fourth rakʿahs of Dhuhr, ʿAṣr, and ʿIshāʾ, it is not Sunnah; after the first two rakʿahs the Sunnah is to restrict the recitation to al-Fātiḥah alone, and if one does recite