Is it permissible to recite the Istikhara supplication immediately after an obligatory prayer, and is it necessary to repeat the prayer if the person performing Istikhara remembers the intention for Istikhara at the end of the Fajr prayer?

The Istikhara supplication should not be recited after an obligatory prayer. Rather, the one performing Istikhara must offer two rak'ahs that are not obligatory, according to the saying of the Prophet, peace and blessings be upon him: "If any of you intends to do a matter, let him pray two rak'ahs other than the obligatory prayers, and then say: 'O Allah, I seek Your guidance by Your knowledge...'"

The thoughts that occurred to you during the prayer do not invalidate it. And if you prayed Fajr with the intention of both the obligatory prayer and Istikhara together, the obligatory prayer is not invalidated. This is because the Istikhara prayer is similar to Tahiyyat al-Masjid (greeting the mosque prayer), and scholars have explicitly stated the validity of the prayer of one who combines the intention of an obligatory prayer with Tahiyyat al-Masjid.