Can we break halacha to avoid embarrassment?

(כ) יְהוּדָה בֶן תֵּימָא אוֹמֵר, הֱוֵי עַז כַּנָּמֵר, וְקַל כַּנֶּשֶׁר, וְרָץ כַּצְּבִי, וְגִבּוֹר כָּאֲרִי לַעֲשׂוֹת רְצוֹן אָבִיךָ שֶׁבַּשָּׁמָיִם.

(20) Yehudah ben Teimah says: Be brazen like the leopard, light like the eagle, swift [running] like the deer, and mighty like the lion to do the Will of your Father Who is in Heaven.

ארבעה טורים א, א Arbaah Turim 1:1

(1) 1. Judah ben Teimah... begins with 1) "strong like a tiger," for it is a great rule in the service of the Divine, as at times one wishes to do some command, and he is prevented from doing so by other people who scoff at him. Therefore be careful to persevere against the scoffers, and don't allow them to impede you from performing a command. Thus R' Yochanan ben Zakkai said to his students, "May it be God's will that your fear of Heaven be as great as your fear of other people." Similarly he said regarding shame, that "sometimes one is ashamed before those who are greater than oneself more than he is ashamed before the blessed Creator." Thus be careful to harden your attitude against the scoffers and do not be ashamed.

(יז) אבל ומנודה שמהלכים בדרך מותרים בנעילת הסנדל וכשיגיעו לעיר יחלוצו וכן בתשעה באב. הגה: וכן במקום שדרים בין הנכרים לא יחלוץ כי אם ברחוב היהודים וכן נהגו (טור והגהות מיימוני והגהות מרדכי ריש פרק קמא):

A mourner is allowed to wear shoes when travelling, and upon arriving at a city must take his shoes off. The same applies on Tisha b'Av. Rema: Similarly if non-Jews live in the area only take off your shoes when in a jewish street.

...כתבו האחרונים דאם צריך לילך לבין הנכרים אעפ"כ אסור לנעול סנדל אף דאיכא למיחש שילעיגו עליו כיון די"א שהוא דאורייתא:

On Yom Kippur, if you walk amongst the non-Jews, it is forbidden to put on shoes even if they will mock you because some say it is a Torah prohibition.

(לג) (לג) מותרים בנעילת הסנדל - וטוב שאז יתן עפר במנעלים:

(לד) (לד) לא יחלוץ וכו' - משום העכו"מ שמלעיגים עלינו:

It is permitted to wear shoes on Tisha beav - however it is a good practise to put soil inside the shoe. and one is permitted to wear shoes because the non-Jews may mock the wearer.

(ה) עֵ֕קֶב אֲשֶׁר־שָׁמַ֥ע אַבְרָהָ֖ם בְּקֹלִ֑י וַיִּשְׁמֹר֙ מִשְׁמַרְתִּ֔י מִצְוֺתַ֖י חֻקּוֹתַ֥י וְתוֹרֹתָֽי׃

(5) inasmuch as Abraham obeyed Me and kept My charge: My commandments, My laws, and My teachings.”

(ה) חקותי דְּבָרִים שֶׁיֵּצֶר הָרָע וְאֻמּוֹת הָעוֹלָם מְשִׁיבִין עֲלֵיהֶם, כְּגוֹן אֲכִילַת חֲזִיר וּלְבִישַׁת שַׁעַטְנֵז, שֶׁאֵין טַעַם בַּדָּבָר, אֶלָּא גְּזֵרַת הַמֶּלֶךְ וְחֻקּוֹתָיו עַל עֲבָדָיו:

(5) חקותי MY ORDINANCES — matters which our evil inclination and the heathen nations argue against the necessity of prohibiting, such as the eating of pork and the wearing of garments made of a mixture of wool and linen — things for which there are no apparent reasons but which are the King’s decrees and enactments imposed on His subjects (Yoma 67b).

How many distinctions can we see which allow or disallow halacha to be broken?

Q: What about shaking hands with a woman?

Q: Wearing a kippah in an antagonistic neghbourhood?

(A) אמר רב יהודה אמר רב,המוצא כלאים בבגדו פושטן אפי' בשוק מ"ט (משלי כא, ל) אין חכמה ואין תבונה ואין עצה לנגד ה' כ"מ שיש חלול השם אין חולקין כבוד לרב...

(B) ת"ש,גדול כבוד הבריות שדוחה [את] לא תעשה שבתורה,ואמאי,לימא,אין חכמה ואין תבונה ואין עצה לנגד ה',תרגמה רב בר שבא קמיה דרב כהנא בלאו (דברים יז, יא) דלא תסור,אחיכו עליה,לאו דלא תסור דאורייתא היא,אמר רב כהנא,גברא רבה אמר מילתא לא תחיכו עליה כל מילי דרבנן אסמכינהו על לאו דלא תסור ומשום כבודו שרו רבנן,

(a) Version #1 (our text) (Rav Yehudah): If one finds Sha'atnez in his garment, even in the market, he must take it off.

(c) Question: What is the reason?

(d) Answer: "Ein Chachmah v'Ein Tevunah v'Ein Etzah l'Neged Hash-m" -- wherever desecration of Hashem's name is involved, we do not give honor to an important person (to leave the garment on him).

1. Kevod ha'Beriyos (human dignity) overrides [any] Lav of the Torah.

(Rav bar Sheva): It overrides [only] the Lav of "Lo Sasur" (do not veer from what the Rabbis say).

(k) Chachamim of Eretz Yisrael laughed at this. If so [Kevod ha'Briyos overrides Chachamim's rulings about Mitzvos], one may transgress any Lav!

(l) Rav Kahana: Do not laugh at anything that a great Chacham says! All mid'Rabanan laws are based on Lo Sasur. Chachamim [pardoned their honor, and] said that Kevod ha'Beriyos overrides their laws.

...How were our predecessors different from us that miracles occurred for them but not for us? ,Our predecessors jeopardised their lives for the sanctification of the Name, but we do not. ,As it once happened with Rab Adda b. Ahabah who noticed a heathen woman wearing a karbalta in the street; thinking that she was an Israelite, he went up to her and tore it from off her., It was then discovered that she was a heathen, and they fined him four hundred Zuzim.,