Recently at a morning minyan at UOS there was a need to give two Yisraeilim an Aliyah (they each had a Yartzheit). Our custom is to understand that in such situtions the Kohanim would be happy to forgo their honor (without having to be asked - see sources below) so those with Yartzheits (or who have other reasons to have an aliyah) can receive an aliyah. Calling a Kohein first, would require calling a Levi second, leaving only one Aliyah for a Yisrael and hence leaving out one of the Yisraelim who had Yartzheit.
That morning, the Gabbai chose to call three Yisraeilim (another option (maybe) would be Levi, Yisrael, Yisrael). A Levi who was present felt that the Gabbai had acted improperly and had disrespected the Levi's status ) Calling Levi , Yisrael, Yisrael would have also made it possible for both Yisraeilim to get their Aliyot. Below is a collection of sources on this issue. As you can see, the Gabbai was correct (acc. to some a Levi should specifically not be called in that situatuion!) in calling three Yisraeilim.
Sources confirming that the Kohanim do not need to 1. be asked to forgo their honor and 2. leave the room in a situation when there are two Yisraeilim who need an aliyah.
Rabbi Yaakov Chaim Sofer (Kaf Hachaim 135: 14) quoting Rav Shlomo Kluger who notes that Ashkenazim in Israel would call a Yisrael intead of a Kohein and the Gabbai would say (withoit requiring the Kohanim to leave) "Af Al Pi SheYeish Kohein" (even though there is a kohein). This was neccesary as there were often visitors to those synagogues who needed to be honored with an Aliyah The assumption was that the kohanim would waive their honor.
Shanah B'shanah is a journal (published by Heichal Shlomo) dedicated to organizing synagogue practices with articles on Halacha, aggadah and other issues. In the (1972-1973, -pg. 130-131) edition, Rabbi Aron Pichnik suggests that, without asking the Kohanim to leave, call up the non Kohein by saying "B'Rishut Hakohanim" - "with the permission of the Kohanim".
Sources on what happens when the regular order (Kohein, Levi, Yisrael) is broken.
אמר אביי נקטינן אין שם כהן נתפרדה חבילה
§ Abaye said that we have a tradition that if there is no priest there in the synagogue at the time of the Torah reading, the bundle is separated, i.e., a Levite is not shown precedence over Israelites.
נתפרדה חבילה -
1. נפסק הקשר איבד הלוי את כבודו בשביל חבילתו הנפרדת ואינו קורא כלל כך אמר מורי הזקן ומורי ר' יצחק בן יהודה וכן סידר רב עמרם אבל מתלמידי מורי רבי יצחק הלוי שמעתי משמו שאין סדר לדבר להקדים לוי לישראל ומי שירצה יקדים:
1. The connection (between the Kohein and the Levi) is therefore the Levi has lost his place of honor and is NOT CALLED AT ALL (Acc. to this view, the Levi his sanctity from his service to the Kohen, and in the absence of a Kohen, the Levi has no special status.)
2. There is no order (when there is no Kohein) requiring a Levi be called first, and the Gabbais can call whoever he wishes.
...ואם אין כהן בב"ה קורא ישראל במקום כהן ולא יעלה אחריו לוי: הגה אבל ראשון יוכל לעלות. וכשקוראין אותו אומרים במקום כהן שלא יטעו לומר שהוא כהן:
(כד) יוכל לעלות - דלא גרע מישראל ודוקא כשאין הישראל גדול מהלוי דאל"ה יקראוהו ראשון ושוב לא יקראו ללוי כלל [אחרונים].
Rav Moshe Feinstein writes that when there is no Kohein, that either Yisrael, Yisrel, Yisrael or Levi, Yisrael, Yisrael are acceptable.
Rabbi Yosef Dov Soloveitchik, a Levi, would often refuse the first aliyah when there was no Kohen present, despite being the greatest talmid chacham present (Rabbi Hershel Schachter, Nefesh HaRav [1994], 130; Rabbi Aharon Ziegler, “Halakhic Positions of Rabbi Joseph Ber Soloveitchik,” Jewish Press, April 7, 2000). He felt that it was improper to ignore the opinion of Rashi that once the “bundle is separated,” the Levi should not receive the first aliyah. The Levi’s honor derives solely from his service to the Kohen. Thus, in the absence of a Kohen, he lacks special standing.
(What’s the Truth about . . . Giving a Levi the First Aliyah? Rabbi Ari Z. Zivatofsky)
שו"ת אגרות משה אורח חיים חלק ג סימן יט
באין כהן ואיכא לוי איך להתנהג...
בדבר המחלוקת בביהכ"נ כשאין כהן בביהכ"נ ואיכא לוי איך צריכין להתנהג. הנה הדין אליבא דהרמ"א שכן נוהגין במדינותינו הוא שאין מחוייבין לקרא להלוי במקום הכהן אבל רשאין לקרותו כשרוצין ראשון במקום כהן, ואם קראו ישראל במקום כהן אזי אין לקרות להלוי מהז' קרואים אלא אחר הז' קרואים לאחרון ולמפטיר. ואין נוהגין אצלינו לעשות חלוק מצד הגדלות כדהביא הט"ז /באו"ח/ בסימן קל"ה סק"ו ואולי הוא מצד מחלוקת וכדומה, לכן נוהגין לקרות למי שירצו ראשון במקום כהן ואם קראו לישראל ראשון אין קורין שוב להלוי אלא לאחרון ומפטיר, ועיין גם במ"ב שם ס"ק כ"ד.
A fascinating incident occurred in 1664 in Germany, as recorded by Rav Yair Chaim Bacharach.21 One Friday evening, a member of the community died, and because the roof of the home containing the body was connected to the shul roof, Kohanim could not enter the shul that Shabbat. The town rabbi, Rav Moshe Shimshon Bacharach, instructed that seven Yisraelim be called up; he did not want to give a Levi the first aliyah. Although it was pointed out to him that the Shulchan Aruch and many others permit giving a Levi the first aliyah, he refused to do so, and insisted that Yisraelim get all the aliyot. Subsequently, he clarified that if a Levi has a chiyuv or is more learned than everyone else present, he may receive the first aliyah.
(What’s the Truth about . . . Giving a Levi the First Aliyah? Rabbi Ari Z. Zivatofsky)
See here for more on thid and related topics.
https://jewishaction.com/religion/jewish-law/whats-truth-giving-levi-first-aliyah/
(סה) בלא רצון - ואין לאדם להתקוטט בעבור שום מצוה
that one should not fight over any mitzvah