To recap your question:
- Are there any halachic problems that would prevent an observant Jew from using someone else's (meaning someone who either identifies as transgender or non-binary, or who has undergone gender reassignment surgery, whether Jewish or non-Jewish) preferred pronouns?
The simple answer is yes, for an observant Jew this type of practice, meaning acceding to their request and using their preferred, personal pronouns, like they do themselves, is prohibited according to halacha.
The prohibition comes from the written Torah, Vayikra 20:23 which says:
וְלֹ֤א תֵֽלְכוּ֙ בְּחֻקֹּ֣ת הַגּ֔וֹי אֲשֶׁר־אֲנִ֥י מְשַׁלֵּ֖חַ מִפְּנֵיכֶ֑ם כִּ֤י אֶת־כׇּל־אֵ֙לֶּה֙ עָשׂ֔וּ וָאָקֻ֖ץ בָּֽם׃
You shall not follow the practices of the nation that I am driving out before you. For it is because they did all these things that I abhorred them
And also from Vayikra 18:3 and 18:26 which say:
כְּמַעֲשֵׂ֧ה אֶֽרֶץ־מִצְרַ֛יִם אֲשֶׁ֥ר יְשַׁבְתֶּם־בָּ֖הּ לֹ֣א תַעֲשׂ֑וּ וּכְמַעֲשֵׂ֣ה אֶֽרֶץ־כְּנַ֡עַן אֲשֶׁ֣ר אֲנִי֩ מֵבִ֨יא אֶתְכֶ֥ם שָׁ֙מָּה֙ לֹ֣א תַעֲשׂ֔וּ וּבְחֻקֹּתֵיהֶ֖ם לֹ֥א תֵלֵֽכוּ׃
You shall not copy the practices of the land of Egypt where you dwelt, nor of the land of Canaan to which I am taking you; nor shall you follow their laws.
וּשְׁמַרְתֶּ֣ם אַתֶּ֗ם אֶת־חֻקֹּתַי֙ וְאֶת־מִשְׁפָּטַ֔י וְלֹ֣א תַעֲשׂ֔וּ מִכֹּ֥ל הַתּוֹעֵבֹ֖ת הָאֵ֑לֶּה הָֽאֶזְרָ֔ח וְהַגֵּ֖ר הַגָּ֥ר בְּתוֹכְכֶֽם׃
But you must keep My laws and My rules, and you must not do any of those abhorrent things, neither the citizen nor the stranger who resides among you;
To understand how this seemingly innocent and unrelated action, to follow the request of the transgender and/or non-binary community, to describe them by their preferred personal pronouns (like the practice which they do themselves) can be connected to these clear prohibitions in the written Torah requires reviewing how these concepts are taught via our Oral law across many different sources.
The first source to review is Sifra (Torat Kohanim, Vienna edition, Acharei Mot, Chapter 13)
The beginning of this chapter emphasizes that both Jews and non-Jews are warned by the Creator of us all regarding prohibited sexual practices. Specific details and consequences may vary depending upon status, but this subject is addressed to everyone.
These prohibitions also apply to both men and to women.
It is no surprise that our generation faces this specific challenge now, at the end of exile and the beginning of the final redemption. It is found in the commentary of the Rogatchover Gaon to the Chumash, parshat Acharei Mot, 18:2 quoting Rabbi Yehuda HaNasi from the Sifra 13:4 which says that G-d knew that in the end Israel would be broken, disrupted and disconnected through prohibited sexual relationships. (לינתק בעריות) That their exposure to this type of sin (during their enslavement and exile) would damage and corrupt them (much like severe Post Traumatic Stress Disorder). And the Rogatchover points explicitly to Sanhedrin 63b and the quote of Rav Yehuda in the name of Rav who says:
Rav Yehuda says that Rav says: The Jewish people knew that idol worship is of no substance; they did not actually believe in it. And they worshipped idols only in order to permit themselves to engage in forbidden sexual relations in public, since most rituals of idol worship would include public displays of forbidden sexual intercourse.
That (exposure to) sin causes sin. And this was actually the advice that Bilaam gave to Balak after failing to curse the Jewish people during the first redemption. His curses being converted into prophecies about the future, final redemption.
In Sifra 13:6-7 it distinguishes the practices specifically of the Egyptians and the Canaanites of that time because they were the most extreme examples of their time in regard to sexually deviant practices. It points out that these types of practices are considered a spiritual pollution of the individual and that this spiritual pollution of the individual also results in the pollution of surrounding physical environment to such an extent that the surrounding environment (literally the land) will no longer support them and will eventually vomit them out! (Vayikra 18:25, 18:28 and 20:22)
If today you look around with alarm at the widespread and varied destruction of the environment on a global scale and the impending signs of a potential mass extinction event, the enormous forest fires caused by acts of G-d making the very air of millions of people unfit to breath, and the multiple volcanoes spewing ash, smoke, lava and sulfur fumes (literally fire and brimstone), it is worth reflecting upon what the Torah teaches is its spiritual cause.
It is very much in keeping with the concept of Divine Judgement Middah-Keneged-Middah (Measure for Measure) but in an inverse relationship to the Source of all life, meaning G-d. Like is emphasized within the Torah in regard to the concept of the first commandment: to be fruitful and multiply, meaning the perpetuation of the species, diversification and increase of life is a primary sign of a healthy and thriving environment. (See Bereshit 1:22 and Bereshit 1:28)
The political justification of the LGBTQ+ movement is all about diversity, but in alignment with the opposite of life.
This is the underlying message from the Torah about the causes behind the mass extinction event recounted in connection with Noach and the flood. See Bereshit 6:1-13 and the commentaries there for details.
Although we have a Heavenly promise that a mass extinction event via water will not be repeated, we do not have a guarantee that we will not experience another mass extinction event by other means, be it by disease or some other catastrophe like earthquake, volcano, extreme climate change or devastating pollution of the environment.
Sifra goes on to explain that Idol Worship, Prohibited Sexual Practices and Murder are all from the same spiritual root. In Sifra 13:9 it also connects this to the concept of passing sons and daughters through the flames of Molach. (Devarim 18:10)
Essentially this is the wasting of ones reproductive potential, which is comprised of both sons and daughters, by indulging irrationally in obsessive excess with the flames of natural sexual desire (this desire is something G-d created for a blessing and for good in connection to the reproduction and preservation of the species, meaning the commandment to Be fruitful and multiply). Like G-d told to Adam HaRishon, he was to do this commandment and to preserve and guard it! (Bereshit 2:15)
In the commentary of the Raavad to Sifra 13:9 he emphasizes that any of the customs and practices like excessive preoccupation with personal grooming and beautification associated with these prohibited sexual behaviors are included in the prohibition against following the ways of the Amorites.
And Rabbi Meir also explains that this activity is associated with the Amorites, who are also connected with the Amalekites and their allies in Sodom and Gomorrah. (Bereshit 14:1-7)
And this prohibition is explained in detail by Yerucham Fishel Perla in his commentary to Saadia Gaon's Sefer HaMitzvot, Negative Commandments, Number 12 and how it includes also their customs and practices. It is not limited to actual worship of a physical idol, but includes their mode of dress, hairstyles and grooming and even unusual behavior. All these types of things vary according to what is practiced in that region and generation.
It is also the view of Rambam as found in his Sefer HaMitvot, Negative Mitzvot 30, and Mishneh Torah, Sefer HaMada, Hilchot Avodah Zarah, 10:1 and for deeper understanding of Rambam's position see Moreh Nevuchim, Section 3:37.
This is also the view found in Sefer Yere'im by the Tosaphist, Rabbi Eliezer of Metz, see Sefer Yere'im with commentary To'afot Romm, Freidman Edition, Vol.3, Amud 7, Siman 271 and Siman 313 who explains that this prohibition pertains to practices that lead to and are in support of men marrying men, women marrying women, women marrying more than one man, and men marrying a woman and her daughter, etc.
ובחוקותיהם לא תלכו לא אמרתי אלא בחוקים החקוקים להם ולאבותיהם מה היו עושים איש נושא איש ואשה נושא אשה אשה נשאת לשנים ואיש נושא אשה ובתה לכך נאמר ובחוקותיהם לא תלכו:
The same view is expressed in the Semag, Sefer Mitzvot Gadolot of Rabbi Moshe of Coucy, Negative Commandments, 50.
It is also the position found in Shulchan Aruch, Yoreh Deah, Siman 178 which says:
One [i.e., a Jew] should not follow the customs of non-Jews (nor should one try to resemble them)
To which the Rema, Rabbi Moshe Isserles emphasizes:
Rather, one [i.e., a Jew] should be distinct from them [i.e., non-Jews] in one’s manner of dress and in all of one’s actions. But all of this [i.e., these restrictions] apply only to things that non-Jews do for the sake of licentiousness.
And a thorough treatment can also be found in the Responsa of the third Lubavitcher Rebbe in Tzemach Tzedek, Section Yoreh Deah, Siman 91.