Is the Fajr prayer, in which the obligatory (fard) part was performed before the sunnah, unacceptable, knowing that the intention was to perform the obligatory part first?

The obligatory prayer is valid, and its validity does not depend on performing the Sunnah before it. However, performing the regular Sunnah prayers, such as the two rak'ahs of Fajr, after the obligatory prayer is contrary to the Sunnah, unless their time was missed before the obligatory prayer due due to time constraints; in which case, they should be made up afterwards, as long as the sun has not risen, or during the forenoon (Duha) time.