Are there any denominations that defend their doctrines by appealing to extra-biblical evidence from the 1st/2nd century of the early Church?

There are a couple of doctrines that appeal to early non-scriptural accounts.

Explanation of Brothers

The Infancy Gospel of James refers to an "old Joseph" and "young Mary". In the story, Joseph is portrayed as a widower with children from a previous marriage.

Subsequent to that, Jerome rejected that explanation and basically invented the cousin's of Jesus to explain the relationship. Jerome believed both Joseph and Mary remained ever-virgin.

Typically, Eastern Orthodox subscribe to the first view, while Roman Catholic subscribe to Jerome's idea.

The Catechism of Catholic Church tries to portray it this way.

500 Against this doctrine [that Jesus was born normally] the objection is sometimes raised that the Bible mentions brothers and sisters of Jesus.157 The Church has always understood these passages as not referring to other children of the Virgin Mary. In fact James and Joseph, "brothers of Jesus", are the sons of another Mary, a disciple of Christ, whom St. Matthew significantly calls "the other Mary". They are close relations of Jesus, according to an Old Testament expression.

Ever-virgin

The same gospel is used to explain that Mary remained a virgin even during the birthing stage. This subsequently was spelled out that Mary was a virgin before, during, and after Jesus' birth.

Here's how Wikipedia explains the same history.

By the 3rd century, the doctrine of the perpetual virginity of Mary had become well established; important early Christian theologians such as Hippolytus (170–235), Eusebius (260/265–339/340) and Epiphanius (c. 310/320–403) defended it. By then the early church did not accept Mary had any children apart from Jesus. Eusebius and Epiphanius held these children were Joseph's children from a previous marriage. Epiphanius adds Joseph became the father of James and his three brothers (Joses, Simeon, Judah) and two sisters (a Salome and a Mary or a Salome and an Anna)[38] with James being the elder sibling. James and his siblings were not children of Mary but were Joseph's children from a previous marriage. Joseph's first wife died; many years later, at the age of eighty, "he took Mary (mother of Jesus)". According to Epiphanius the Scriptures call them "brothers of the Lord" to confound their opponents.[39][40] Origen (184–254) also wrote "according to the Gospel of Peter the brethren of Jesus were sons of Joseph by a former wife, whom he married before Mary".

The apocryphal History of Joseph the Carpenter, written in the 5th century and framed as a biography of Joseph dictated by Jesus, describes how Joseph had with his first wife four sons and two daughters. His sons' names were Judas, Justus, James, and Simon, and the names of the two daughters were Assia and Lydia. Years after his first wife died, he took Mary. Therefore, the brothers of Jesus would be the children of Joseph by his first wife.

The Protoevangelium of James explicitly claims Joseph was a widower, with children, at the time that Mary is entrusted to his care.

The Catholic Encyclopedia, citing the texts contained in the apocryphal writings, writes that:

When forty years of age, Joseph married a woman called Melcha or Escha by some, Salome by others; they lived forty-nine years together and had six children, two daughters and four sons, the youngest of whom was James (the Less, "the Lord's brother"). A year after his wife's death, as the priests announced through Judea that they wished to find in the tribe of Juda a respectable man to espouse Mary, then twelve to fourteen years of age. Joseph, who was at the time ninety years old, went up to Jerusalem among the candidates; a miracle manifested the choice God had made of Joseph, and two years later the Annunciation took place.

Jerome (c. 347–420), another important early theologian, also held the perpetual virginity doctrine, but argued these adelphoi were sons of Mary's sister, whom Jerome identified as Mary of Cleopas. The Oxford Dictionary of the Christian Church mentions a modern scholar, whom it does not identify, has proposed these men were the sons of Clopas (Joseph's brother according to Hegesippus) and of Mary, the wife of Cleopas (not necessarily referring to Jesus' mother's sister).

On the other hand, claiming that the brothers of Jesus were in fact step-brothers (same mother, different fathers), there is this evidence.

Sextus Julius Africanus's reference to "desposyni" (blood relatives of Jesus related to his nuclear family) is preserved in Eusebius of Caesarea's Ecclesiastical History:

For the relatives of our Lord according to the flesh, whether with the desire of boasting or simply wishing to state the fact, in either case truly, have handed down the following account.

This view obviously means that Mary would not have remained in the child-birth state and did not remain an ever-virgin.

Easter Question as it relates to Col 2:16-17

Another early dispute was on how to observe the death, burial, resurrection of Jesus Christ. Asia Minor observed the time following the Jewish calculation as floating days based on the timing of the sun and moon. Those in Rome wanted to observe it on Sundays only.

1. A question of no small importance arose at that time. For the parishes of all Asia, as from an older tradition, held that the fourteenth day of the moon, on which day the Jews were commanded to sacrifice the lamb, should be observed as the feast of the Saviour’s passover.1687 It was therefore necessary to end their fast on that day, whatever day of the week it should happen to be. But it was not the custom of the churches in the rest of the world to end it at this time, as they observed the practice which, from apostolic tradition, has prevailed to the present time, of terminating the fast on no other day than on that of the resurrection of our Saviour.

1. But the bishops of Asia, led by Polycrates, decided to hold to the old custom handed down to them.1695 He himself, in a letter which he addressed to Victor and the church of Rome, set forth in the following words the tradition which had come down to him:1696 2. “We observe the exact day; neither adding, nor taking away. -Eusebius Church History-

So, in the Catholic Catechism:

1166 "By a tradition handed down from the apostles which took its origin from the very day of Christ's Resurrection, the Church celebrates the Paschal mystery every seventh day, which day is appropriately called the Lord's Day or Sunday."36 The day of Christ's Resurrection is both the first day of the week, the memorial of the first day of creation, and the "eighth day," on which Christ after his "rest" on the great sabbath inaugurates the "day that the Lord has made," the "day that knows no evening." The Lord's Supper is its center, for there the whole community of the faithful encounters the risen Lord who invites them to his banquet:

As regards Colossians 2:16 (Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath days:), we just read that the Easter dispute really wasn't about observing the feasts per se, but was about either observing Passover as a floating period but always on the 14th (like the 25th of Christmas) or was it about trying to always observe the Sunday appearances.

The issue of Colossians 2:16 was about those who continued to teach the necessity of observing the Law to attain salvation. As Peter put it in Acts 15:11:

But we [Christians} believe that through the grace of the Lord Jesus Christ we [Jews] shall be saved, even as they [the Gentiles].

Adding works to grace has been a non-starter from the beginnings of Christianity, but certain denominations continue to teach the necessity of more in order to be saved.

And you, being dead in your sins and the uncircumcision of your flesh, hath he quickened together with him, having forgiven you all trespasses; Col 2:13

The confessional writings of the Lutheran church appeal to the role of early church tradition in helping to remove obstacles for the proper interpretation of Scripture.

For example, at the end of the Augsburg Confession, it states (alternate translation in Latin):

... in doctrine and ceremonies nothing has been received on our part against Scripture or the Church Catholic. For it is manifest that we have taken most diligent care that no new and ungodly doctrine should creep into our churches.

... in doctrina ac ceremoniis apud nos nihil esse receptum contra Scripturam aut ecclesiale catholicam, quia manifestum est, nos diligentissime cavisse, ne qua nova et impia dogmata in ecclesias nostras serperent.

In the Apology to the Augsburg Confession the following is also stated:


... But we cheerfully maintain the old traditions made in the Church for the sake of usefulness and tranquility; and we interpret them in a more moderate way ... (Article XV)

Also, in the Lutheran Confessions there is a section called the Catalogue of Testimonies that contains a number of quotes from the early church fathers.

Having read Ken Graham’s excellent answer in the question asking if any of the early church fathers believed they had to follow dietary laws, one would think that the Adventists defend their dietary regiment by appealing to extra-biblical evidence from the 1st and 2nd centuries of the early church.