You raise an excellent question. I believe it is best if we answer it within the context of the term “Church”. A Church is a community of believers headed by a bishop. To head a local Church, the bishop must be in communion with the Bishop of Rome.
That being said, since the local diocese is a real Church and its head is a real successor to the Apostles, because he’s a bishop. A bishop does not need permission to grant or deny faculties to anyone in his diocese. He makes that move motu proprio, his own decree. There is no right of appeal over the head of a local bishop. The Bishop of Rome cannot overrule a local bishop unless there is evidence of wrong-doing. That would have to be a violation of Church law, civil law, heresy or apostasy.
The only place where a bishop does not have authority is in a house of religious men. The major superior grants the faculties or he may allow the local superior to grant them.
Since the SSPX are not religious, they do not have Pontifical Rights as do Dominicans, Franciscans, Trappists, Benedictines, Carmelites or Jesuits.
JR