My family is all Sunni, very few of us are followers of the Ahlulbayt, but not openly.
If a person has missed Fajr prayers and still should make-up for them, it is possible for him to join the Tarawih prayers during the month of Ramadan with such intention. His prayers would count as Qada prayers and not Tarawih. However keep in mind that congregational Tarawih prayers as we see nowadays in Sunni masjids were not performed by the Prophet in that fashion.
With prayers for your success.