In answer to the question in the OP, take a look at this verse from Deuteronomy 22:
The woman shall not wear that which pertaineth unto a man, neither shall a man put on a woman's garment: for all that do so are abomination unto the LORD thy God.
Does it say that only Hebrews who do this are abomination unto the LORD? Does it say only those who did so 3000 years ago were an abomination? No, it doesn't say that. It doesn't put a time or cultural limitation on it.
If God finds it an abomination, it shouldn't bother us to much if some of us find it to be an abomination/repulsive/disgusting. I find it rather disgusting, in particular, if a man is dressed up in drag trying to be an attractive woman. It's also very self-destructive, too. I forget the suicide rates on those who go all the way to the extreme of transgendered operations. I think it was around 40%. I read an article from someone from Johns Hopskins who helped develop the perverted procedure who said their hospital got out of it when they found out how destructive these operations were.
The ancient world had their versions of transgenders, eunuchs who dressed up like women. (They probably weren't as artistic about chopping them up back then, but it's the same thing. Post-op transgendered men are eunuchs who dress like women.) They had people who engaged in homosexual activity. Philo, in the first century wrote about them. He thought transvestites, transgenders, those who performed homosexual acts, etc. all deserved to die.
The Canaanites had plenty of sexual perversion. We think of religion as promoting at the least decent morals. Back then, their religion didn't. They had fertility rituals of all types, and sexual immorality as a part of it. Polytheism is often that way. Greeks had an idol for drinking and so they could drink and make it a religious festival, too. Hinduism is the same way. People used to sacrifice to Kali, and there is a goddess for prostitutes in Hinduism. Any nasty sin has a god attached to it for polytheists.
They used to mix sexual perversion with idolatry. But that doesn't make sexual immorality okay if we divorce it from paganism. First century pagans mixed paganism with prostitution, but hiring atheist prostitutes doesn't make prostitution okay. Prostitution is forbidden and idolatry is forbidden. Paul teaches against both issues separately in I Corinthians. So we can't say prostitution is wrong only because of idolatry.
And we shouldn't say that cross dressing was wrong only because we guess that it was mixed with paganism like so many aspects of Canaanite daily life were. If all who do such are an abomination to the LORD, then those who do so without a religious motivation are as well.