In my research and search for the origins of Biblical scripture I have been digging in and looking for source material from which all Christian translations are based.  I finally found a website that provides a lot of information on the Torah, Talmud and many more,  https://www.sefaria.org/texts.  When searching for the meaning of keli and crossdressing this is what I found (https://www.sefaria.org/Nazir.59a.2?lang=bi)

The Gemara asks: And what does the first tanna, who holds that the prohibition is by rabbinic law, learn from this verse: “A man shall not put on a woman’s garment”? The Gemara answers: He requires it for that which is taught in the baraita, where it states: “A woman shall not wear that which pertains to a man, and a man shall not put on a woman’s garment, for whoever does these things is an abomination to the Lord your God” (Deuteronomy 22:5). What is the meaning when the verse states this? If it teaches only that a man may not put on a woman’s garment, and a woman may not wear a man’s garment, it is already stated in explanation of this prohibition that “it is an abomination to the Lord your God,” and there is no abomination here in the mere act of wearing a garment.

אֶלָּא שֶׁלֹּא יִלְבַּשׁ אִישׁ שִׂמְלַת אִשָּׁה וְיֵשֵׁב בֵּין הַנָּשִׁים, וְאִשָּׁה שִׂמְלַת אִישׁ וְתֵשֵׁב בֵּין הָאֲנָשִׁים. רַבִּי אֱלִיעֶזֶר בֶּן יַעֲקֹב אוֹמֵר: מִנַּיִן שֶׁלֹּא תֵּצֵא אִשָּׁה בִּכְלֵי זַיִין לְמִלְחָמָה — תַּלְמוּד לוֹמַר: ״לֹא יִהְיֶה כְלִי גֶבֶר עַל אִשָּׁה וְלֹא יִלְבַּשׁ גֶּבֶר שִׂמְלַת אִשָּׁה״, שֶׁלֹּא יִתַּקֵּן אִישׁ בְּתִיקּוּנֵי אִשָּׁה.

Rather, it means that a man may not wear a woman’s garment and thereby go and sit among the women; and a woman may not wear a man’s garment and sit among the men. Rabbi Eliezer ben Ya’akov says: From where is it derived that a woman may not go out with weapons to war? The verse states: “A woman shall not wear that which pertains to a man, and a man shall not put on a woman’s garment,” which indicates that a man may not adorn himself with the cosmetics and ornaments of a woman, and similarly a woman may not go out with weapons to war, as those are for the use of males. Rabbi Yoḥanan’s ruling follows this opinion.

This support what I have to what Paul was referencing in Deuteronomy 22:5, especially when it comes to women donning armor and taking up arms in order to go to war.  In the time of Paul, crossdressing for the purpose of avoiding was or entering a circle of women was forbidden so the idea of men wearing cosmetics (see Genesis 42:1-26 when Joseph was in the court of the Egyptian King, He wore make up).  It was a man's duty to fight and go to war and men dressing as women to avoid fight might have had a lot to do with Paul's message.  Again, I refer to an earlier post about Joan of Arc doing just that, donning a soldiers armor, taking up weapons of war and fighting in the 100 years war.  The Catholic Church made her a saint after her death, so wouldn't that go against Deuteronomy 22:5.

Thought I would post this before I lost it or forgot to do so.