Must I go to the mosque to pray the missed Fajr prayer at nine in the morning, or should I pray it at home, knowing my eagerness to perform prayers in the mosque?

If you miss congregational prayer due to unintentional sleep, there is no sin upon you. You must pray it as soon as you wake up, because that is its time in your regard. You are not obligated to go to the mosque, and you can pray at home, but it is better to perform it in congregation if possible, whether in the mosque with other worshippers (even if they are performing supererogatory prayers), or at home with someone who prays with you.