It's a good question and actually has an interesting answer.
The reason why this particular detail is important and can't be changed is because it relates to two of the fundamental Principles of Jewish faith, the coming of Moshiach and the resurrection of the dead.
From the 10th to the 22nd lines are all describing and/or relating to what is mentioned in the 9th line, namely the two great sources of light which G-d created. This is the paradigm which G-d established on the fourth day of creation as stated in Bereshit 1:14-15 which says:
וַיֹּ֣אמֶר אֱלֹהִ֗ים יְהִ֤י מְאֹרֹת֙ בִּרְקִ֣יעַ הַשָּׁמַ֔יִם לְהַבְדִּ֕יל בֵּ֥ין הַיּ֖וֹם וּבֵ֣ין הַלָּ֑יְלָה וְהָי֤וּ לְאֹתֹת֙ וּלְמ֣וֹעֲדִ֔ים וּלְיָמִ֖ים וְשָׁנִֽים׃
וְהָי֤וּ לִמְאוֹרֹת֙ בִּרְקִ֣יעַ הַשָּׁמַ֔יִם לְהָאִ֖יר עַל־הָאָ֑רֶץ וַֽיְהִי־כֵֽן׃
So the key is understanding what is being referred to with the two sources of light.
These sources of light are referring to Moshiach ben Yosef, the Moshiach of Exile, which is compared to night and Moshiach ben David, the Moshiach of the redemption, which is compared to day.
This relationship is also alluded to from Esther 8:16 which connects the idea of light to joy and gladness.
לַיְּהוּדִ֕ים הָֽיְתָ֥ה אוֹרָ֖ה וְשִׂמְחָ֑ה וְשָׂשֹׂ֖ן וִיקָֽר׃
And also from the words of the Prophet of the Redemption, Isaiah in 35:10
וּפְדוּיֵ֨י יְהוָ֜ה יְשֻׁב֗וּן וּבָ֤אוּ צִיּוֹן֙ בְּרִנָּ֔ה וְשִׂמְחַ֥ת עוֹלָ֖ם עַל־רֹאשָׁ֑ם שָׂשׂ֤וֹן וְשִׂמְחָה֙ יַשִּׂ֔יגוּ וְנָ֖סוּ יָג֥וֹן וַאֲנָחָֽה׃
This follows what is taught in the name of the Vilna Gaon in Kol HaTor 2:147.
, Isaiah 51:3
כי־נחם יהוה ציון נחם כל־חרבתיה וישם מדברה כעדן וערבתה כגן־יהוה ששון ושמחה ימצא בה תודה וקול זמרה
and Isaiah 51:11.
ופדויי יהוה ישובון ובאו ציון ברנה ושמחת עולם על־ראשם ששון ושמחה ישיגון נסו יגון ואנחה
The roots of Sassone (gladness) and Simchah (joy) are שש and שמח, which begin the letter Sin which sounds like the letter Samech and actually replaces that letter in the order of the Aleph-Beit of this piyut.
That Samach (שמח) is connected with Moshiach ben David is actually alluded to by G-d directly in Shemot 4:13-14 which says:
ויאמר בי אדני שלח־נא ביד־תשלח
ויחר־אף יהוה במשה ויאמר הלא אהרן אחיך הלוי ידעתי כי־דבר ידבר הוא וגם הנה־הוא יצא לקראתך וראך ושמח בלבו
Just as שלח־נא ביד־תשלח refers to Moshiach, the final Redeemer, so too ושמח בלבו also refers to Moshiach.
And this also alludes to one of the details of the final redemption, namely the elimination of death (סם מות) which comes to the world via the Angel of death, Sama'el spelled סמאל.
And that relates to the conversion of the letter Samech to the letter Sin, in the same type of sign that was given to Moshe Rabbeinu in the first redemption as mentioned in Shemot 4:17 and Shemot 7:8-15.
Moshe was told to demonstrate to Pharaoh, to show his staff, one of the signs of G-d's Kingship, the King's scepter, being converted into the Serpent (the same Serpent (נחש) described in the Garden of Eden and connected with the introduction of death into the world in Bereshit 3:1-6) and then converted back again into Moshe's staff. And this relates to part of what happens in the final redemption with Moshiach, the elimination of death via the resurrection of the dead.
That in fact, the word שמח itself, which is associated to Moshiach ben David, also alludes to this relationship because the At-Bash transformation of the letter Chet (ח) is the letter Samech (ס). And this means the letters Shin and Samech are exact opposites in relation to the letter Mem of שמ״ח, meaning שמ״ס. And this שךדם alludes to the Gematria (numerical) equivalency of נחש (Nachash/Serpent/meaning the Angel of Death) and משיח (Moshiach).
