What is the ruling on performing the Sunnah of Fajr immediately after concluding the Fajr prayer, due to an inability to maintain ablution and a fear of it being invalidated before the Fajr prayer, and to avoid embarrassment when re-performing ablution?

It is permissible, according to some scholars, to delay the pre-Fard (obligatory prayer) Sunnah (supererogatory) prayers until after the Fard prayer, even though it is better to offer them before. The time for the Sunnah prayers that precede the Fard prayers begins with the start of the Fard prayer's time and remains as long as the Fard prayer's time has not expired. Consequently, it is permissible to delay the pre-Fajr Sunnah prayer until after the Fajr obligatory prayer, and there is no sin in doing so, even if the delay is intentional.