Zimmun is the introduction to Birkas Hamazon. We invite those present to join in the bentching (ex. “Rabosai mir velen bentchen”). Why do we do this after we have already begun the recitation of bentching, with the prefacing passages of “Al Naharos Bavel” on weekdays or “Shir Hamaalos” on Shabbos? It would make more sense to make such an invitation before we say these passages, rather than after.
The answer to this question is that Shir Hamaalot/Al Naharot Bavel are not "prefacing passages" to Birkat Hamazon, but are actually supposed to be recited during the meal itself, prior to beginning Birkat Hamazon.
As noted by the Shelah (Sha'ar Ha'otiyot, Kedushat Ha'achilah 170; some cite it as page 82b, such as here), the root of this practice is the Zohar's statement (Parshat Terumah, Volume 2, page 157b; some cite it as 158b or 154b) that one who is enjoying a meal should take time to recall the Holy Land and the destruction of the Temple, and what the reward for doing so is:
מאן דאתעדן על פתוריה ומתענג באינון מיכלין אית ליה לאדכרא ולדאגא על קדושה דארעא קדישא ועל היכלא דמלכא דקא אתחריב, ובגין ההוא עציבו דאיהו קא מתעצב על פתוריה בההוא חדוה ומשתיא דתמן קודשא בריך הוא חשיב עליה כאלו בנה ביתיה ובנה כל אינון חרבי דבי מקדשא זכאה חולקיה
This works well with various other customs to remember the destruction of the Temple at celebratory or enjoyable times, but should not be confused with being a part of Birkat Hamazon itself.
For some halachic sources to support this, Magen Avraham (Orach Chaim 1:5, citing the aforementioned passage from Shelah) brings this halacha on the Shulchan Aruch's comment that one who fears Heaven should remember and mourn over the destruction of the Beit Hamikdash, and not in the laws of Birkat Hamazon. Additionally, Olat Tamid (Orach Chaim 180:2, also citing Shelah) writes that one should wash Mayim Acharonim only after Shir Hama'alot and Al Naharot Bavel.
Therefore, it is clear that one should recite Shir Hama'alot or Al Naharot Bavel first, and only once the meal is finished should they invite others to participate in Birkat Hamazon by reciting the Zimmun.