What is the ruling on someone who prays behind the Imam, but cannot complete Al-Fatiha before the Imam bows due to a speech impediment? Should he bow with the Imam or pray alone at home?

Congregational prayer is an individual obligation, and it is not permissible to abandon it except for a valid excuse. Your slow recitation is not an excuse; rather, it is obligatory to attend the congregational prayer and to quicken your recitation. If the Imam bows before you complete Al-Fatiha, continue reciting it until you finish it, and this is an excuse that permits you to fall behind the Imam.