In what sense was Mary born without original sin in Roman Catholic dogma?
The simple & direct answer can be found in the Dogma of Immaculate Conception.
But there was a problem in understanding the dogmatic definition as it did not specifically defined when does the merit of Christ was applied. We are not alone in this difficulty as even the Angelic Doctor St.Thomas Aquinas cannot fathom when in the following references;when was the "sanctification" begin?
ST. THOMAS AND THE IMMACULATE CONCEPTION (EDITORIAL NOTE)
1.Whether the Blessed Virgin was sanctified before her birth from the womb?
2.Whether the Blessed Virgin was sanctified before animation?
3.Whether the Blessed Virgin was cleansed from the infection of the fomes?
4.Whether by being sanctified in the womb the Blessed Virgin was preserved from all actual sin?
5.Whether, by her sanctification in the womb, the Blessed Virgin received the fullness of grace?
6.Whether after Christ, it was proper to the Blessed Virgin to be sanctified in the womb?
In all the six articles above it implied that "sanctification in the womb" will removed or cleanse one soul of the original sin contracted at the moment of conception. But the Church teaches that Mary unlike Jeremiah & John the baptist was not conceived in sin.
"O Mary conceived without sin, pray for us who have recourse to thee"(Mary's revelation to St.Catherine Laboure).https://en.wikipedia.org/wiki/Miraculous_Medal
If Mary was conceived without sin, it means She does not need "sanctification in the womb" of St.Anne as the definition of "orginal sin" was only a privation of sanctifying grace. Since Mary was conceived without sin the rational soul coming from God was not deprived of sanctifying grace as there's no barrier according to Church definition.
So, where does the merit of Christ applied to Mary in order to remove the original sin inherited by all mankind thru Adam's fault?
The Dogma of immaculate Conception speaks of Jesus and Mary as the New Adam & New Eve meaning they are created not in the order of nature but in the order of grace. Jesus humanity was assumed and his existence was not by virtue of human seed but of Divine Origin coming from the Abba Father.
496 From the first formulations of her faith, the Church has confessed that Jesus was conceived solely by the power of the Holy Spirit in the womb of the Virgin Mary, affirming also the corporeal aspect of this event: Jesus was conceived "by the Holy Spirit without human seed".146 The Fathers see in the virginal conception the sign that it truly was the Son of God who came in a humanity like our own. Thus St. Ignatius of Antioch at the beginning of the second century says:
You are firmly convinced about our Lord, who is truly of the race of David according to the flesh, Son of God according to the will and power of God, truly born of a virgin,. . . he was truly nailed to a tree for us in his flesh under Pontius Pilate. . . he truly suffered, as he is also truly risen.147
How about Mary? she comes from the human seed of St.Joachim., therefore if the revelation of Mary to St.Catherine Laboure is that "she was conceived without sin" it only follows that "St.Joachim human seed" was not corrupted and the "marital act" between St.Joachim and St.Anne was also devoid of lust or concupiscence. This is a mystery.
The Dogma of Immaculate Conception teaches Mary was pure & immaculate meaning no blemish or stain.
It is written (Canticles 4:7): "Thou art all fair, O my love, and there is not a spot in thee!" But the fomes implies a blemish, at any rate in the flesh. Therefore the fomes was not in the Blessed Virgin.https://isidore.co/aquinas/summa/TP/TP027.html#TPQ27OUTP1
The Dogma of Immaculate Conception teaches the purity of Mary's flesh meaning her human nature was not in fallen or wounded state like all mankind;
Of a Super Eminent Sanctity
To these praises they have added very noble words. Speaking of the conception of the Virgin, they testified that nature yielded to grace and, unable to go on, stood trembling. The Virgin Mother of God would not be conceived by Anna before grace would bear its fruits; it was proper that she be conceived as the first-born, by whom “the first-born of every creature” would be conceived. They testified, too, that the flesh of the Virgin, although derived from Adam, did not contract the stains of Adam, and that on this account the most Blessed Virgin was the tabernacle created by God himself and formed by the Holy Spirit, truly a work in royal purple, adorned and woven with gold, which that new Beseleel made. They affirmed that the same Virgin is, and is deservedly, the first and especial work of God, escaping the fiery arrows the the evil one; that she is beautiful by nature and entirely free from all stain; that at her Immaculate Conception she came into the world all radiant like the dawn. For it was certainly not fitting that this vessel of election should be wounded by the common injuries, since she, differing so much from the others, had only nature in common with them, not sin. In fact, it was quite fitting that, as the Only-Begotten has a Father in heaven, whom the Seraphim extol as thrice holy, so he should have a Mother on earth who would never be without the splendor of holiness.
The question is how can Mary's flesh or human nature coming from the seed of St.Joachim and generated thru the marital act of St.Joachim and St.Anne be transmitted free from original sin and all stain of sins.
The answer is. the merit of Christ Redemptive Sacrifice was applied in St.Joachim as he is the carrier of the "human seed" as revealed to Blessed Catherine Emmerich and the "marital act" was devoid of lust or concupiscence as revealed to St.Bridget
This Church approved private revelation or visions of Blessed Catherine Emmerich and St.Bridget supported the 8th century teaching of the Eastern Church Father a Doctor of the Church in the name of St.John Damascene;https://www.ccel.org/ccel/emmerich/lifemary.v.html
“O most blessed loins of Joachim from which came forth a spotless seed! O glorious womb of Anne in which a most holy offspring grew.” John of Damascus, Homily I (ante A.D. 749).
The Dogma of Immaculate Conception teaches eternal enmity between the Woman and the Serpent and it start with the "seed". Eternal enmity or absolute enmity means Mary was not touch by satan's influence beginning from the "seed or the unformed body".
Psalm 139:16 New International Version (NIV)
16 Your eyes saw my unformed body;
all the days ordained for me were written in your book
before one of them came to be.
This psalmist identifies an "unformed body" in contrast to Jeremiah's "formed" body(Jeremiah1:5) and even to Book of Genesis narration as it stated "God formed Adam". Does this passage refers to Mary's existence in the form of "human seed" alone not yet united to the egg cell of the Woman?
Interpreters of the Sacred Scripture
The Fathers and writers of the Church, well versed in the heavenly Scriptures, had nothing more at heart than to vie with one another in preaching and teaching in many wonderful ways the Virgin’s supreme sanctity, dignity, and immunity from all stain of sin, and her renowned victory over the most foul enemy of the human race. This they did in the books they wrote to explain the Scriptures, to vindicate the dogmas, and to instruct the faithful. These ecclesiastical writers in quoting the words by which at the beginning of the world God announced his merciful remedies prepared for the regeneration of mankind — words by which he crushed the audacity of the deceitful serpent and wondrously raised up the hope of our race, saying, “I will put enmities between you and the woman, between your seed and her seed” — taught that by this divine prophecy the merciful Redeemer of mankind, Jesus Christ, the only begotten Son of God, was clearly foretold: That his most Blessed Mother, the Virgin Mary, was prophetically indicated; and, at the same time, the very enmity of both against the evil one was significantly expressed. Hence, just as Christ, the Mediator between God and man, assumed human nature, blotted the handwriting of the decree that stood against us, and fastened it triumphantly to the cross, so the most holy Virgin, united with him by a most intimate and indissoluble bond, was, with him and through him, eternally at enmity with the evil serpent, and most completely triumphed over him, and thus crushed his head with her immaculate foot.http://www.papalencyclicals.net/Pius09/p9ineff.htm
So, in the O.P question referring to John Henry Newman;
"Does he [the objector] not believe that St. John Baptist had the grace of God—i.e., was regenerated, even before his birth? What do we believe of Mary, but that grace was given her at a still earlier period? All we say is, that grace was given her from the first moment of her existence.
What do we believe of Mary, but that grace was given her at a still earlier period?
The answer is Yes, the graces and the merit of Christ Redemption was given earlier not in the womb of Anne according to St.Catherine Laboure but it was given and applied to the St.Joachim the pious Father of the Blessed Virgin Mary as the scriptures testified;
"Children's childre;6n are the crown of old men: and the glory of children are their fathers. (Proverbs17)
Jesus humanity was assumed and the seed does not comes from a human father but thru the Divine Abba Father, while the seed of Mary comes from a pious human father who was cleansed by the merit of Christ Redemption out pouring a "spotless seed".
Indeed the human nature of Mary comes from pure, virgin and immaculate clay untouch by satan;But this does not mean that Mary no longer need redemption, the Church teaches in CCC492 that "Mary was redeemed in a more exalted fashion" and that definition reveals that Mary was redeemed at the very seed of his human father.
“She is born like the cherubim, she who is of a pure, immaculate clay.” Theotokos of Livias, Panegyric for the feast of the Assumption, 5:6 (ante A.D. 650). https://www.scripturecatholic.com/blessed-virgin-mary/#II_Marys_Immaculate_Conception
The Dogma of the Immaculate Conception as defined;
“We declare, pronounce, and define that the doctrine which holds that the most Blessed Virgin Mary, in the first instance of her conception, by a singular grace and privilege granted by Almighty God, in view of the merits of Jesus Christ, the Savior of the human race, was preserved free from all stain of original sin, is a doctrine revealed by God and therefore to be believed firmly and constantly by all the faithful.”
The word "in the first instance of her conception", does not pertain to the conception in the womb of St.Anne but the conception in the "Eternal Thought" of the Abba Father in eternity. St.Maximillian Kolbe's teaching on Mary as the "Created Immaculate Conception" speaks of Mary conception in eternity as CCC492. The book of Fr.Michael Gaitley "33 Days to Morning Glory" expounded the mystery behind St.Bernadette revelation on Mary as "Created Immaculate Conception" and the Holy Spirit as the "Uncreated Immaculate Conception."
492 The "splendor of an entirely unique holiness" by which Mary is "enriched from the first instant of her conception" comes wholly from Christ: she is "redeemed, in a more exalted fashion, by reason of the merits of her Son".136 The Father blessed Mary more than any other created person "in Christ with every spiritual blessing in the heavenly places" and chose her "in Christ before the foundation of the world, to be holy and blameless before him in love".137
To believe that Mary was sanctified in the womb of St.Anne before or even after animation would implied that Mary's human nature was in the fallen state therefore not pure,virgin & immaculate and this teaching will directly contradict the upheld teachings of the Church Fathers written in the Dogma of Immaculate Conception about the spotless purity of the Blessed Virgin Mary
To believe that Mary's humanity comes from the corrupted seed of St.Joachim will also means Her flesh was not immaculate.
To believe that the "marital act" of St.Joachim & St.Anne was tainted by lust or concupiscence meaning Mary's existence was touch by the malice of Satan runs contrary to the Dogma of Immaculate Conception in uphelding "eternal enmity" and also by the Council of Trent definition on the transmission of orginal sin.
To believe that the "first instance of conception" refers to the womb of St.Anne and not the conception in the Heavenly Realm pointing to the "conception of the Eternal Thought of God in eternity" is not in harmony with the Dogma of Immaculate Conception teaching "eternal enmity", St.Maximillian Kolbe teaching on "Created Immaculate Conception" and St.Catherine of Laboure revelation that Mary does not need "sanctification in the womb of St.Anne".
All this mystery was illuminated by St.Catherine of Laboure, Mary was not conceived in sin,and affirmed by revelation of the Blessed Virgin Mary herself to St.Bernadette Soubirous saying "I AM the Immaculate Conception". Therefore, Mary was formed by the Holy Spirit in the order of grace a "new creature"(http://www.vatican.va/archive/ccc_css/archive/catechism/p122a3p2.htm) and specifically stated "that she be wholly borne by God's grace." as wriiten in CCC490.
"O Mary conceived without sin, pray for us who have recourse to thee."