Is it permissible for a worshiper to perform the Fajr prayer in congregation with his family, intending it as a supererogatory prayer for himself, while it is an obligatory prayer for his family, and then go to the mosque to perform the obligatory prayer again with the congregation, or is there another way to reconcile performing prayer in the mosque with his eagerness for his family's prayer, knowing that he is of the Hanafi school of thought?

What you are doing is correct, God willing. Your family's following your lead is an obligation for them, while for you it is a supererogatory prayer. This is sound according to the most authoritative opinions of the scholars, who cite the action of Mu'adh (may Allah be pleased with him) when he used to pray with the Prophet (peace and blessings be upon him) and then return to lead his people in that same prayer. It is important to ensure that this supererogatory prayer is the Sunnah of Fajr, and not a general supererogatory prayer. It is permissible for you to pray the obligatory prayer in the mosque and then lead your household in prayer.