If a בחור took off the door of his principle’s office before Purim, and now the Yeshiva is trying to figure out who did it, and the only one who knows is a בחור who, to begin with, didn’t really like that boy who took off the door. DO you know of any היתר for the ראש הישיבה to ask that boy? (Whatever the answer is, could you please write where you got that from, and who says that). Rabbi Moshe Feinstein תשובה And the שבט הלוי are talking about a case that that boy they are asking who took off the door is not an enemy, the whole question was if it’s good חינוך to do that, on that we have a disagreement רב משה and others, but in a case that he does not like that boy they never spoke about that.
ב"ה Shalom I cannot refer directly to the case in question since this is an educational issue in an institution where I assume there is a Rosh Yeshiva with educational experience, knowledge of Halacha and has taken all the different aspects of the issue into consideration. Taking a stand, would mean to challenge the decision of a Rosh Yeshiva to a situation of which I have no direct knowledge whatsoever. However, in a theoretical situation if it is correct to ask a student to tell of the actions of another student, is an issue in dispute between Rav Moshe Feinstein zt"l and Rav Shternbuch shlit"a as you noted. שו"ת אגרו"ת משה יו"ד ח"ב קג, ח"ד סימן ל, תשובות והנהגות ח"א תתלט) However, Rav Moshe zt'l, does mention the issue of reporting it for pure reasons "lishma", and that in his opinion that would only be applicable when a student came forth on his own to report a negative action, but not when asked to do so. Even where the Chafetz Chaim zt"l (בכלל ד' בבאר-מים-חיים אות ל"ב). allowed a student to come forward to tell of the actions of another student so that others will be wary of him and keep a distance, when he is the sole witness, he must a student trusted by the teacher as he were two witnesses. There are additional 5 conditions enumerated there by the Chafetz Chaim. (See also סימן ע"א עשה לך רב / חלק א') The intentions of Yoseph Hatzaddik to speak about his brothers before his father, were pure, but we still see the terrible results which came about. So considerations of asking one fellow student to tell on another, must be dealt with cautiously or possibly not at all. All the Best.