Is it permissible for someone who wronged his cousin, then repented and falsely swore binding oaths on several occasions to avoid scandal and severed family ties, to expiate these oaths by feeding ten needy people at once?

The aforementioned oath is a false oath (yamin ghamoos), and according to the majority of scholars, it has no expiation except repentance, while the Shafi'is made expiation obligatory for it. A single expiation suffices you even if the oath was repeated, and the sin is removed by repentance as long as you have repented to Allah. Your motive was to unite the family, maintain kinship ties, and cover up a scandal, and denying the accusation without an oath would have sufficed you.