Has the Lord handed over the souls of the redeemed to St Michael?

No, your translation is correct.

While I know of no tradition about this specifically (nor can find one), there is a possible Biblical basis for the concept (the Bible is technically a part of, and not separate from, tradition). We read in Revelation 12 of the spiritual war in heaven between the devil and "his angels," and Michael and "his angels." It would seem from this alone than Michael is the 'head' of the good angels, whereas the devil is the 'head' of the fallen angels. At least as long as this war endures.

As such, it may be safe to assume that in the same way the spirits of the damned are placed under the dominion of the devil, so the souls of the saints are 'in the hands' of Michael, too, inasmuch as angels are over men in authority, and in any case, of course only God is able to place souls under the authority of others anyway. After all, the parallelism in Revelation 12 is an overt allusion to the "enmity" of Genesis 3, between the woman and her seed, and the devil and his seed ("And the dragon was angry against the woman: and went to make war with the rest of her seed"). If we conclude that the saints, whether human or angel (saints applies to both in Scripture - to angels before humans, interestingly), are on the 'same team' in the cosmic, ages-old spiritual war, then we may assume that Michael is set by God the 'commander' of the 'good team,' and Satan the 'bad team.'

Has the Lord handed over the souls of the redeemed to St Michael?

Tibi trádidit Dóminus ánimas redemptórum in supérna felicitáte locamdas.

Here follows the Church’s translation taken from the Exorcismus in Satanam et Angelos Apostaticos (page 40) of Rituale Romanum:

To thee the Lord has entrusted the service of leading the souls of the redeemed into heavenly blessedness.

This passage does reflect an ancient Catholic tradition that St. Michael leads souls to heaven at the moment of death or as they leave purgatory.

In Roman Catholic teachings, Saint Michael has four main roles or offices. His first role is the leader of the Army of God and the leader of heaven's forces in their triumph over the powers of hell.[31] He is viewed as the angelic model for the virtues of the spiritual warrior, with the conflict against evil at times viewed as the battle within.

The second and third roles of Michael in Catholic teachings deal with death. In his second role, Michael is the angel of death, carrying the souls of all the deceased to heaven. In this role Michael descends at the hour of death, and gives each soul the chance to redeem itself before passing; thus consternating the devil and his minions. Catholic prayers often refer to this role of Michael.

In his third role, he weighs souls in his perfectly balanced scales. For this reason, Michael is often depicted holding scales.

In his fourth role, Saint Michael, the special patron of the Chosen People in the Old Testament, is also the guardian of the Church. Saint Michael was revered by the military orders of knights during the Middle Ages. The names of villages around the Bay of Biscay express that history. This role also was why he was considered the patron saint of a number of cities and countries. - Archangel Michael (Wikipedia)

The tradition that St. Michael leads souls to heaven is centuries old.

This is an ancient tradition, in accord with a belief that St. Michael was the angel who weighed the deeds of a deceased person and led them into eternal life. For this reason St. Michael is frequently depicted in statues or paintings holding scales. The liturgy of the Church appears to confirm this role of St. Michael with the following offertory chant said during a Requiem Mass.

O Lord Jesus Christ, King of glory,deliver the souls of all the faithful departed … let the standard-bearer holy Michaellead them into that holy light.

A medieval text known as the Golden Legend relates an encounter Mary had with an angel prior to her death, preparing her for her assumption.

Then behold, an angel stood before her amid a great light and greeted her reverently as the mother of his Lord. “Hail, blessed Mary!” he said, “receive the blessing of him who bestowed salvation on Jacob. See, Lady, I have brought you a palm branch from paradise, and you are to have it carried before your bier. Three days from now you will be assumed from the body, because your Son is waiting for you, his venerable mother.”

The angel’s name is not immediately revealed, but later on when Mary “died,” Jesus comes into the narrative and gives instructions to the holy archangel.

Christ nodded his consent and immediately Michael the Archangel came forward and presented Mary’s soul before the Lord. Then the Savior spoke and said: “Arise, my dear one, my dove, tabernacle of glory, vessel of life, heavenly temple! As you never knew the stain of sin through carnal intercourse, so you shall never suffer dissolution of the flesh in the tomb.” Thereupon Mary’s soul entered her body, and she came forth glorious from the monument and was assumed into the heavenly bridal chamber, a great multitude of angels keeping her company.

While these legendary stories likely have no historical basis, they do provide for us a piece of meditation and invite us to consider how the Virgin Mary was welcomed into heaven by her Son and a company of angels. In particular, it would make sense for the “Prince of All Angels” to be at the forefront of those who welcomed the “Queen of Angels” into heaven. - How St. Michael the Archangel led the Blessed Virgin Mary to heaven

Deacon Guadalupe Rodriguez from Laredo, Texas puts this tradition into clear perspective:

Tradition tells us that St. Michael is the “Guardian of Purgatory” and that he holds “The Keys of Purgatory.” In the life of mystics St. Michael is often seen sending his angels to console souls in purgatory by sprinkling them with the Precious Blood. St. Michael is also seen escorting souls from purgatory into heaven. At other times like on First Saturday, Our Lady comes to release souls from purgatory. Three great saints and doctors of the Catholic Church also concur St. Michael’s intercession and consolation of the souls of purgatory.

“It is incontestably recognized since the foundation of Christianity that the souls of the faithful departed are delivered from Purgatory through the intercession of St. Michael the Archangel.” - St. Robert Bellarmine

“The Prince of the Heavenly militia is all-powerful in Purgatory, and he can assist the poor souls whom the justice and sanctity of the Almighty retain in this place of punishment.” - St. Anselm

St. Michael has received the care of consoling and helping the Souls in Purgatory.” - St. Alphonsus Liguori

This also aligns with one of the early church writings called Apocalypse of Paul which explains how a soul was lifted up in a vision into the heavenly abodes:

“Behold! A river with its waters. I say to the angel, ‘what is it?’ he answers, ‘if anyone is impure or unholy, but repentant, once he has left his body behind, he is led forward first to adore God, and then by the command of the Lord he is handed over to the Angel Michael, who baptizes him in the river and then leads him to the city of God.’” - (Apoc., Paul 22.)

In the book St. Michael and the Angels testimonies from holy monks explain how one holy monk offered mass for a friend recommending his soul to St. Michael and how the priest was able to see the soul escorted from purgatory into heaven by St. Michael the Archangel. In another testimony a priest offers mass also in honor of St. Michael for several souls but adds the prayer, “May the Prince of Angels, St Michael, lead them into the glory of Heaven,” and he is able to see St. Michael descend from heaven into purgatory to deliver the souls. - St. Michael & All Souls