Indeed it is consistent on this matter.
Unfortunately for your position, GWB, the early history of the Church is consistent with the Covenantal view of the matter.
In the middle of the second century Infant Baptism is mentioned not as an innovation, but as a rite instituted by the apostles. Nowhere do we find it prohibited and everywhere we find it practiced. Early in the nascent Church we have St. Irenaeus (c. 130-c. 200) who provides a very early witness to Infant Baptism, based on John 3:5. Irenaeus wrote, "For He [Jesus] came to save all through means of Himself-all, I say, who through Him are born again to God,-infants, and children, and boys, and youths, and old men" (Against Heresies, 2, 22, 4).
Origen (AD c. 185-c. 254) who had traveled to the extents of the Roman Empire wrote with confidence, "The Church received from the Apostles the tradition [custom] of giving Baptism even to infants. For the Apostles, to whom were committed the secrets of divine mysteries, knew that there is in everyone the innate stains of sin, which must be washed away through water and the Spirit" (Commentary on Romans 5, 9). Infant Baptism?, Steve Ray
From the beginning of New Testament Christianity at the Feast of Pentecost (Acts 2: 38-39) to our time, unbroken and uninterrupted; the church has baptized babies. Entire households (Jewish, proselytes and Gentiles) were baptized by Christs original 12 Apostles (I Corinthians 1: 16; Acts 11: 14, 16: 15, 33, 18: 8) and that practice has continued with each generation.
Polycarp (69-155), a disciple of the Apostle John, was baptized as an infant. This enabled him to say at his martyrdom. "Eighty and six years have I served the Lord Christ" (Martyrdom of Polycarp 9: 3). Justin Martyr (100 - 166) of the next generation states about the year 150, "Many, both men and women, who have been Christs disciples since childhood, remain pure at the age of sixty or seventy years" (Apology 1: 15). Further, in his Dialog with Trypho the Jew, Justin Martyr states that Baptism is the circumcision of the New Testament. Irenaeus (130 - 200), some 35 years later in 185, writes in Against Heresies II 22: 4 that Jesus "came to save all through means of Himself - all. I say, who through him are born again to God - infants and children, boys and youth, and old men." Similar expressions are found in succeeding generations by Origen (185 - 254) and Cyprian (215 - 258) who reflect the consensus voiced at the Council of Carthage in 254. -- Infant Baptism in Early Church Historyby Dennis Kastens
The Covenant is Visible and One.
This date is not without some difficulties, as while the Anabaptist Communion is undoubtedly of very great antiquity ("The Baptists are the only body of known Christians that have never symbolized with Rome. ~~ Sir Isaac Newton), precise dating of the formation of the Communion is rendered difficult by the fact that the Anabaptist Communion, wherever it has been found, has always upheld the doctrines of Independence and Separationism.
The doctrine of Local Independence has always ensured that the Elders of any particular AnaBaptist church are independent of any servitude to any higher echelon of Elders (though Biblically, even the local AnaBaptist church is rightly governed by a session of multiple Elders, not a single Elder which is to say, that the proper practice of local congregational governance ought be presbyterian, not simply pastoral, in form, as much for Baptist congregations as for Presbyterian congregations); and the doctrine of Separationism has always required of AnaBaptist congregations that they have no fellowship with the unfruitful works of darkness (Ephesians 5: 11), and accept no communion with those who pervert the Gospel.
So assigning a precise date to the formation of the AnaBaptist Communion is rendered difficult by the fact that this Communion has always been a loose, non-hierarchical grouping of independent congregations. But if Cardinal Hosius (president of the Council of Trent) estimate of twelve centuries of AnaBaptist Persecution be accepted as accurate
This would date the existence of the AnaBaptist Communion to at least AD 350.
Nonetheless, the date of AD 253-254, of the Council of Carthage, enjoys certain historical advantages to recommend it, as Baptism was certainly among the major issues pronounced upon by the Council.
Ah, but theres the rub the determination of this early Council of Carthage was not upon the question of infant Baptism and believers Baptism, but rather upon the question between two different modes of the Infant Baptism which was the ancient and established sacramental practice of the Church, even at that early date.
The judgment of the assembled 66 bishops at Carthage in AD 253-254 was simply this: that Baptism, which required no blood-letting, did not require a delay until the eighth day after birth for the Sacrament to be performed. The propriety of baptizing the infant children of Believers into the Covenant was assumed as fact; the only question was whether or not the Levitical necessity of delay until the eighth day continued as a sacramental regulation of the seal. As no blood was shed, and thus no danger to the infant was involved, the judgment of Carthage was that no continuation of the Levitical delay was required.
The development of the AnaBaptistic practice of with-holding Baptism from Infants was not even considered among the diverse modes of Baptism found among the Churches of that day.
The Covenant is Visible and One.