No, sorry.
The SSPX was founded as a Pious Union under the 1917 Code. Under the 1983 Code, and upon recognization they would become a Society of Apostolic Life.
Canonically speaking, as well-intentioned as Archbishop Lefebvre was, the SSPX 'Third Order' does not and will never exist, it cannot exist! A religious order (not congregation or society) needs to have a First and Second Order in order to have a Third Order. Only the great, major orders have them. The following are the only existing Third Orders:
Augustinian
Benedictine (not a Third Order properly speaking, but rather Oblates of individual houses)
Carmelite
Discalced Carmelite
Dominican
Franciscan
Mercedarian
Minim
Norbertine
Servite
Trinitarian
This is why many pre-Vatican II congregations had associated confraternities and societies but not Third Orders. For example the Redemptorists have the Confraternity of Our Mother of Perpetual Help and St. Alphonsus. The Jesuits the Apostleship of Prayer and the Happy Death Society.
As far as Motu Proprio groups (Societies of Apostolic Life), the FSSP has the Confraternity of St. Peter. The Institute of Christ the King has the Society of the Sacred Heart. The Fraternity of St. Vincent Ferrer has an associated Confraternity of the Rosary. On the sedevacantist side, the CMRI has the Confraternity of Mary Immaculate Queen.
Actually, peterp raises a good question. Why claim to establish a third order of the Franciscans rather than the SSPX-MC?
The third order SSPX was at least based off of being a canonically founded religious order, I think.