In one of Fr. Ripperger's videos, he mentioned that during an exorcism, the demon challenger him as to whose jurisdiction he was under. He named the bishop he was under and the demon ended its challenge and departed.
I can't imagine what it must be like to be an exorcist, but based on this incident, I think it made Fr Ripperger realize the importance of being part of the hierarchy of the Church.
I think, if an SSPX priest were performing the exorcism, the answer would be
"THE CHURCH OF JESUS CHRIST", because in our present day, Rome has lost
the Faith. Therefore the eternal Church is able to provide jurisdiction in cases
of emergency. And we are in emergency times.
This is a concept that Fr Ripperger does not understand, it seems.