How did the early church view transgenderism?

First, the idea of "transgender" and all the current definitions and versions of it from cross-dressing to sex reassignment surgery have been around since antiquity. Many online articles describe individuals and sub-cultures that were more or less accepted in their societies for their alternative lifestyles. There is also a difference between a person's gender view of themself (notice the politically correct non-gender specific reflexive pronoun I just had to force into my dictionary) and their idea of what kind of gender role they are attracted to in another person.

This does not make the practice approved of in the eye of the God of the Bible though. God clearly condemns all non-classical binary gender definitions and pairings as pointed out in the other answers, and Paul does speak against the idea of cross-dressing for whatever reason.

Among all the LGBTQIA+ alternative gender types and lifestyles popular today, there is one that deserves possible acceptance, or at least sympathy and compassion. All except the "I" group describe people that have used their minds to make determinations about how to see and use their bodies, there is no hard science that supports the idea they were born that way.

The "I" group, meaning "Intersex", are people BORN with either both gender sets of genitalia or underdeveloped reproductive systems, which corrupts how their bodies produce or utilize sex hormones. This makes it difficult for them to find a role in society that they and others can be comfortable with. They have been heavily scrutinized for generations when trying to compete as females in athletics. Until very recently they were always forced into a specific classical gender group by an authority, they were never allowed to choose on their own.

The Bible describes the idea of a "eunuch" (meaning a male that has no functioning reproductive system, so he also has a corrupted hormone balance). Jesus himself states in Matthew 19:12 that there are three types of eunuchs that have an acceptable reason to not marry (so also not for a sinful reason). A eunuch could be born that way (intersex), or castrated by another for a specific purpose (likely against their will, to serve royalty or some other group), or voluntarily (possibly virtually but maybe physically) become one to serve God without the distractions of marriage or erotic desire (later in 1 Corinthians 7:1-9 Paul affirms that it is good to avoid marriage to serve God if the temptation to commit fornication can be overcome).

All of these eunuch types were accepted by God. None of these categories include the currently popular idea of a person intentionally corrupting their body for their own satisfaction or entertainment, so that would be an iniquity (willful sin). As an iniquity it could be forgiven by God if the person genuinely repents from their choice and attempts to undo the change and live as their birth gender. Possibly a child who has been deceived by adults into a forced chemical or physical castration would not be guilty of sinning even though it doesn't match up well to Jesus' reason for it.

Before Jesus defined His acceptable reasons for it, the Old Testament in Deuteronomy 23:1 decreed that no castrated male (meaning after birth, does not apply to the intersex) for any reason could "enter the assembly of the LORD", which could either mean get married or maybe join a Godly community.

As for women voluntarily attempting to convert their bodies into an approximation of a male's, this could be an interpretation of the punishment God assigned to Eve and all future women in Genesis 3:16. Part of the verse says that the woman would "desire" her husband but he will be master over her. The definition of "desire" is not a wanting to be with or enjoy the company of. It is the same word that God warns Cain about when He says that sin is at his door and "desires" him (Genesis 4:7), it wants to own and corrupt his nature. This definition of "desire" could possibly mean that the woman wants to take over the man's role (and even body in extreme cases) but she will suffer the frustration of always being subjugated by him. Certainly this would explain a lot of the more militant types of feminism in the free world today. God absolutely approves of women with strong roles in society (consider Lydia as a merchant with a household of employees that served Christ and His early followers) but that is not the same as replacing men in those roles. So there are no acceptable "reverse-eunuch" type models for a person born female but trying to become male to follow, it is all iniquity (but also forgivable with TRUE repentance the same as with eunuchs).

I assume that "transgenderism" can refer to a variety of phenomena from cross-dressing to expressing one's sexuality in a "non-binary" or "abnormal" fashion. Especially, it can refer to same-sex attraction based on gender dysphoria, which would be thought of as h*m*sexuality or "unnatural lust" in ancient times. Before moving to the Church Fathers let's look at the New Testament writers.

NT Writers

The earliest opinions seem come from the letters of Paul.

Here, Paul addresses the issue of dressing inappropriately to one's sex:

Every man who prays or prophesies with his head covered dishonors his head, but every wife who prays or prophesies with her head uncovered dishonors her head, since it is the same as if her head were shaven... Judge for yourselves: is it proper for a wife to pray to God with her head uncovered? Does not nature itself teach you that if a man wears long hair it is a disgrace for him, but if a woman has long hair, it is her glory? For her hair is given to her for a covering. (1. Cor 11:3-15)

Thus, Paul assigns definite gender roles based on clothing, and by implication opposes cross-dressing or even the unintentional appearance of it. On the other hand, he would not have felt the need to write this to the Corinthian church unless others in the church practiced otherwise. We cannot say for certain that Paul's attitude here represented that of the early church.

Paul is even harsher in his condemnation of women and men experiencing what he called 'unnatural lusts' such as would be the case with transgender women feeling same-sex attraction because they are a man trapped in a woman's body, or vice versa.

God gave them over to shameful lusts. Even their women exchanged natural sexual relations for unnatural ones. In the same way the men also abandoned natural relations with women and were inflamed with lust for one another. Men committed shameful acts with other men, and received in themselves the due penalty for their error. (Rom. 1 26-32)

The Letter of Jude likewise condemns abnormal sexuality. It apparently has h*m*sexuality in mind, but since h*m*sexual and lesbian acts can be expressions of transgenderism, this scripture should also be included:

Sodom and Gomorrah and the surrounding towns gave themselves up to sexual immorality and perversion. They serve as an example of those who suffer the punishment of eternal fire. (Jude 1:7)

Early Church Fathers

Among the Church Fathers, a notable passage is found in Justin Martyr, who bemoaned the birth of numerous hermaphrodites and also spoke of a type of sex-change operation, which he denounced as a "mutilation."

For this pollution [prostitution] a multitude of females and hermaphrodites, and those who commit unmentionable iniquities, are found in every nation. . . And there are some who prostitute even their own children and wives, and some are openly mutilated for the purpose of sodomy; and they refer these mysteries to the mother of the gods” (First Apology 27 [A.D. 151]).

The Didache declared: “You shall not commit murder, you shall not commit adultery, you shall not commit pederasty (Didache 2:2).

The Exhortation to the Greeks, attributed to Clement of Alexandra, blesses a Scythian king mentioned by Herodotus because the king killed one of his citizens who:

having been deprived of his own virility in Greece, was now communicating the effeminate disease to his fellow Scythians.

Tertullian denounced all kinds of non-marital and non-binary sexual behaviors:

Whoever enjoys any other than nuptial intercourse, in whatever place, and in the person of whatever woman, makes himself guilty of adultery and fornication... But all the other frenzies of passions — impious both toward the bodies and toward the sexes — beyond the laws of nature, we banish not only from the threshold, but from all shelter of the Church, because they are not sins, but monstrosities. (On Modesty ch. 4)

Finally, Cyprian of Carthage denounces men who tend toward effeminacy to the point that the man (whether literally or not) "becomes a woman":

Men are emasculated, and all the pride and vigor of their sex is effeminated in the disgrace of their enervated body; and he is more pleasing there who has most completely broken down the man into the woman. He grows into praise by virtue of his crime; and the more he is degraded, the more skillful he is considered to be. Such a one is looked upon—oh shame!—and looked upon with pleasure. . . (Letters 1:8 [A.D. 253]).

The online article What the Early Church Believed: Homosexuality presents a more complete survey of the Church Fathers' attitudes toward h*m*sexuality and transgenderism.

Regarding the question about whether the Fathers interpreted the cited verses differently from "us," this depend on which modern readers are referred to in the OP. None of the Church Fathers or NT writers overtly approved of transgenderism and to the extent that they addressed the practice, they denounced it. However this does not mean they would take the view that certain types of transgenderism [especially natural hermaphroditism] should be excluded from the Church. Tertullian may be an exception. In addition Paul himself wrote that "in Christ there is no male or female." This saying may be used by inclusivist Christians today to support the view that transgenderism should be welcomed unconditionally in the church, but it hardly represents any known attitude of the NT writers or Church Fathers.