Our Gemara on Amud Aleph tells us that the modest Cohanim would not enter the fray to grab a portion of the Lechem Hapanim, despite it being a mitzvah to eat. On a practical halakhic level, this teaching is used as a source and reminder that it does not bring honor to a mitzvah if you fight over it, as we can see in Baer Heytev OC 53:27.
However, is this teaching so clear-cut? The Gemara (Yoma 39a) provides more detail to the story:
In the times when Shimon Hatzaddik served as High Priest, all kinds of miracles and blessings prevailed, including:
וְנִשְׁתַּלְּחָה בְּרָכָה בָּעוֹמֶר, וּבִשְׁתֵּי הַלֶּחֶם, וּבְלֶחֶם הַפָּנִים. וְכׇל כֹּהֵן שֶׁמַּגִּיעוֹ כְּזַיִת, יֵשׁ אוֹכְלוֹ וְשָׂבֵעַ, וְיֵשׁ אוֹכְלוֹ וּמוֹתִיר. מִכָּאן וְאֵילָךְ, נִשְׁתַּלְּחָה מְאֵירָה בָּעוֹמֶר וּבִשְׁתֵּי הַלֶּחֶם וּבְלֶחֶם הַפָּנִים, וְכׇל כֹּהֵן מַגִּיעוֹ כְּפוּל. הַצְּנוּעִין מוֹשְׁכִין אֶת יְדֵיהֶן, וְהַגַּרְגְּרָנִין נוֹטְלִין וְאוֹכְלִין. וּמַעֲשֶׂה בְּאֶחָד שֶׁנָּטַל חֶלְקוֹ וְחֵלֶק חֲבֵירוֹ, וְהָיוּ קוֹרִין אוֹתוֹ "בֶּן"
A blessing was sent upon the offering of the omer; and to the offering of the two loaves from the new wheat, which was sacrificed on Shavuos; and to the shewbread, which was placed on the table in the Temple. Due to that blessing, each priest that received an olive-bulk of them, there were those who ate it and were satisfied, and there were those who ate only a part of it and left over the rest because they were already satisfied from such a small amount. From then onward, a curse was sent upon the omer, and to the two loaves, and to the shewbread, that there were not sufficient quantities to give each priest a full measure. Therefore, each priest received just an amount the size of a bean; the discreet, pious ones would withdraw their hands since a bean-bulk was less than the quantity needed to properly fulfill the mitzva, and only the voracious ones would take and eat it.
Rashi (ibid) says that the modest ones held back from grabbing the bread only after the blessings ceased. This implies that even if there was a possibility of some fighting, as long as there was still a blessing in the bread, the modest Cohanim did not withdraw from the fray. Shall we then conclude, according to Rashi, that it is worth causing a l fight over a mitzvah? Machatzis HaShekel (OC 53:26) resolves these questions with the following approach: When there is a complete mitzvah and only a possibility of strife, one should not abstain. When there is only a partial mitzvah (such as Lechem Hapanim that is of a small unsatisfying portion) and a possibility of strife, one should surely abstain. When there is a sure mitzvah, AND a certainty of strife, Machatzis HaShekel is not sure if one should abstain or jump into the fray.
These discussions have obvious implications for those who are obligated to say Kaddish and similar situations. No matter how important it is to bring a zechus to the niftar, the greatest zechus is no fighting. Keeping a mitzvah pure is important, as we shall see soon in the Psychology of the Daf Kiddushin 56.
Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation 
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