Are Catholics permitted to honour Mary as a priest?
The answer is Yes, all Catholics are permitted. St. Pius X even granted 300 days indulgences on the faithfuls who will invoke Mary's intercession under this title.
"Mary Virgin Priest, pray for us". (St. Pius X; 1906)
Controversy arises during reign of Pope Benedict XVI, that led to the Holy See to forbid the propagation of images of the Mary wearing the vestment of the priest, but it's clear that the Holy See does not forbid the saying of the prayer devotion.
Pope Benedict personally addressed in numerous letters the pilgrims at Marian sanctuaries. He named Mary the Patron of Bavaria, and permitted, in Mexico, the Feast of the Immaculate Conception of Guadaloupe. He authorised the Feast of Mary Mediator of all Graces. He condemned the misuse of Marian statues and pictures, dressed in priestly robes, which he outlawed 4 April 1916. (AAS 1916 146 Baumann in Marienkunde; 673) - Pope Benedict XV (Wikipedia)
What Pope Benedict XV condemned was not the prayer devotion instituted by St. Pius X but the misuse of the images.
So, the prayer devotion honoring and imploring the intercession of Mary Virgin Priest continues until today, unless a Pope officially prohibits the acts of St. Pius X pertains to the devotion and to granting indulgences.
Is Mary Virgin Priest a real priesthood? The answer is Yes. A Pope cannot grant indulgences to a metaphorical priest. See this link on the basis of St. Pius X on approving the prayer devotion to Mary Virgin Priest.
What is the basis of St. Pius X in 1906 that led to the approval of the prayer devotion; "Mary Virgin Priest, pray for us"?
Granting that Mary is a Virgin Priest, the question is when and who ordained Mary into priesthood? Is her priesthood an ordinary priesthood like the presbyters or a full priesthood like the Bishop?
Certainly Mary, was not yet a priest during her childhood days even if it's spent serving in the Temple for ten years. And after that, we can see from the gospel that Mary was betrothed to a man name Joseph. This is a mystery, so where can we find the first mystery?
The Holy Rosary starts with the Joyful Mystery and it is the "Mystery of the Annunciation".
There exists the belief that the Annunciation acted as Mary’s ordination. There is also a belief that through her ‘yes’, Mary made Christ present in the world, in her womb – as the priest makes Christ present in the words of consecration. In an address by Bishop Nazlian, at Lourdes in 1914 said “ A priest has the power to mystically produce the body of the Lord giving that body its sacramental form . . . . I allow myself to say that Mary is the first to say Mass, by agreeing to the Incarnation and so preparing the victim.” - The Priesthood of Mary
Does the Annunciation ordained Mary into priesthood? Let's ponder the gospel passage from the heart like Mary ponder and treasure everything from her heart.
And the angel being come in, said unto her: Hail, full of grace, the Lord is with thee: blessed art thou among women. (Luke1:28 ; Douay-Rheims)
- The angel pronounce the word "Hail", it is a salutation appropriate only for a King & Queen. Therefore, the angel acknowledge Mary as Queen.
What does a "greeting" gives? When Mary visited her cousin Elizabeth her "greetings" made Elizabeth and her baby leaped for joy and be "filled with the Holy Spirit".
The angel greetings also, brings the outpouring of the gifts of the Holy Spirit it made Mary's state of soul not just filled by grace but rather it was "full of grace".
Is this what happen to all the Apostles in the Upper Room they were all "filled with the Holy Spirit"? Yes.
"All of them were filled with the Holy Spirit and began to speak in other tongues[a] as the Spirit enabled them."(Acts2:4)
God said in Amos3:7;
" For the Lord God doth nothing without revealing his secret to his servants the prophets."
- King David was anointed by prophet Samuel for him to become Prince of Israel. Eventually became King of Israel.
Samuel Anoints David
- Jesus Baptism in the Jordan river, the coming down of the Holy Spirit was in the presence & witnessed by the Last Great Prophet St.John the Baptist.
The Baptism of Jesus
- St.Paul was "filled by the Holy Spirit" before he was sent in the mission in the presence of the prophets.
Now in the church at Antioch there were prophets and teachers: Barnabas, Simeon called Niger, Lucius of Cyrene, Manaen (who had been brought up with Herod the tetrarch) and Saul. 2 While they were worshiping the Lord and fasting, the Holy Spirit said, “Set apart for me Barnabas and Saul for the work to which I have called them.” 3 So after they had fasted and prayed, they placed their hands on them and sent them off. (Acts13:1-3)
- The Apostles received the anointing of the Holy Spirit in the presence of Mary the Prophet. Is Mary a prophet, Yes.
She is the Queen of Prophets. See the the Litanies Lauretane
Is Archangel Gabriel a servant of God? Gabriel reveals himself to Zechariah,
“I am Gabriel,” replied the angel. “I stand in the presence of God, and I have been sent to speak to you and to bring you this good news. "(Luke1:19)
Is Gabriel a prophet, since he comes from the Heavenly Realm, he is not only a prophet but a celestial prophet in the Kingdom of God.
Did Archangel Gabriel prophesy to Mary? Yes!
"You will conceive and give birth to a son, and you are to call him Jesus. 32 He will be great and will be called the Son of the Most High. The Lord God will give him the throne of his father David, 33 and he will reign over Jacob’s descendants forever; his kingdom will never end.” (Luke1:31-32)
So, God is the one who sent his servant Archangel Gabriel who prophesy to Mary His Divine Plan for the Logos.
What is the rank of angel Gabriel in the Kingdom of God?
Gabriel (/ˈɡeɪbriəl/; Hebrew: גַּבְרִיאֵל, lit. 'Gavri'el "God is my strength"', Ancient Greek: Γαβριήλ, lit. 'Gabriel', Coptic: Ⲅⲁⲃⲣⲓⲏⲗ, Aramaic: ܓܒܪܝܝܠ, Arabic: جبريل, Jibrīl or جبرائيل Jibrāʾīl), in the Abrahamic religions, is an archangel. He was first described in the Hebrew Bible and was subsequently adopted by other traditions.
In the Hebrew Bible, Gabriel appears to the prophet Daniel to explain his visions (Daniel 8:15–26, 9:21–27). The archangel appears in such other ancient Jewish writings as the Book of Enoch. Alongside archangel Michael, Gabriel is described as the guardian angel of Israel, defending this people against the angels of the other nations.
The Gospel of Luke relates the stories of the Annunciation, in which the angel Gabriel appears to Zechariah and the Virgin Mary, foretelling the births of John the Baptist and Jesus, respectively (Luke 1:11–38). Many Christian traditions—including Anglicanism, Eastern Orthodoxy, and Roman Catholicism—revere Gabriel as a saint.
Islam regards Gabriel as an archangel sent by God to various prophets, among them Muhammad. The first five verses of the 96th chapter of the Quran, the Clot, is believed by Muslims to have been the first verses revealed by Gabriel to Muhammad.Archangel Gabriel (Wikipedia)
The Archangel also said "the Lord is with you" signifying the presence of the "spiritual mark" of the priesthood like the "holy chrism" which put a "spiritual mark" on the priesthood of a priest or bishops.
And lastly, the Angel Gabriel finished ordination by pronouncing the blessings saying "blessed are you among women".
Did Mary share in the Priesthood of her Son? De Salazar, a great Marian theologian of the 17th century says of Mary, “not only did Christ pour the fullness of his priestly annointing on Mary but He emptied it on her” and therefore of her role at Calvary; “ the Blessed Virgin fulfilled the office of the Priest in that showing her will in all things to conform to the will of her son, she offered and sacrificed him on the altar of the Cross as Christ did himself.”
What is the proof of Mary has the faculties of a priest to ordained and confer the gifts of the Holy Spirit?
Vincent de Paul in the 17th Century, says of the Visitation to Elizabeth, “The Blessed Virgin, as Bishop in the Church, sanctified the son of the high priest Zachariah. She sanctified St. John and through the imposition of her power, using her right as Mother of God and spouse of the Father, she imprinted the Holy Spirit on St. John”. - The Priesthood of Mary
In closing, Mary Virgin Priest is an honor given to Mary permitted by the Church and granted an indulgences to all the faithful who seek Mary's intercession under this title.
Mary's priesthood is a real priesthood as St.Vincent De Paul affirmed that Mary's Visitation to St.Elizabeth sanctified St.John the Baptist and She imprinted the Holy Spirit on his soul as "spiritual mark".
The only question is, if Mary was not ordained to a Sacramental Order of priesthood, what is the order of Her ordination?