Is it permissible for an Imam to repeat Surat Al-Fatihah or some of its verses multiple times with different recitations in a single rak'ah before reciting what is easy of the Qur'an?

Repeating Al-Fatihah in prayer is not legislated and is disliked according to the four madhhabs, but the prayer is not invalidated by it. The Hanafis have made it obligatory to perform the prostration of forgetfulness (sujud as-sahw) for whoever repeats it, unless he recited a surah and then repeated Al-Fatihah. However, if the repetition is due to obsessive whispering (waswasah) or a disability, there is no dislike in it. Repeating some verses of Al-Fatihah is disliked without need and does not invalidate the prayer. The Malikis hold that whoever repeats Al-Fatihah forgetfully, whether it was certain or doubtful, must perform the prostration of forgetfulness. However, if one repeats it intentionally, there is no prostration, and the preferred view is that the prayer is not invalidated, though one incurs sin.