The FSSP is completely in union with Rome. They were founded in 1988 by Bl. John Paul II.
It’s founding was led by a group of priests from the SSPX who no longer wanted association with that group due to disobedience.
This is a quote from their declaration in 1988.
“For the honour and glory of the holy Catholic Church, for the consolation of the much troubled faithful, and for the peace of their conscience, the undersigned, members until now of the Fraternity of Saint Pius X, declare with profound regret over the illicit consecration of bishops on 30 June [1988] that they have remained within the Catholic Church as pars sanior of this same Fraternity, and that they have but one desire: to be able to live as a religious society in this Church and place themselves at her service under the authority, of course, of the Roman Pontiff, her supreme head.”