A. Interesting question actually. Here’s some background for those who might not be familiar: The famous Kabbalist Rabbi Isaac Luria, best known as the “Arizal” composed three Kabbalistic songs for each of the Shabbos meals. “Azameir B’Shvachin” is for Friday Night, “Asader L’Seudosah” (which the Rebbe taught a lively melody for) is for Shabbos Lunch, and “Bnei Heichalah” is for the third meal Seudah Shlisht or Shalosh-Seudos near the end of Shabbos. There are Chabad Niggunim melodies to all three Arizal songs, but the Niggun of the 3rd meal is associated with the Alter Rebbe and is best known.
Chabad (which closely follows the Arizal in terms of prayer rites as well as Kabbalistic/Chassidic perspective and philosophy) has all three of these Arizal songs in its Siddur and Bentschers, but most other communities only have the Bnei-Heichalah for the third Shabbat meal and not the other two. Why not?
Here’s a sociological answer. Not for sure, but it’s an idea. What makes the third Shabbos meal different than the first two, is that in many communities (aside for college campus Chabads etc) the first two Shabbat meals are eaten at home with the family, while the third meal is usually at the synagogue between the Mincha and Maariv prayer. It’s likely that a tradition such as the Arizal’s songs would have been better preserved as part of a synagogue ritual than at mealtime at home.
Remember that many of the early Jewish Eastern European immigrants who settled in America were from Chassidic background. In those days, sadly, most didn’t keep observant for very long, there was a lot of assimilation. Back then, there were many Nusach Ari or Nusach Sfard (i.e. Chassidic, not Sephardic) synagogues all over the country, especially the smaller synagogues established by Polish and Russian Jews. Less and less of them kept their traditions, especially at the first half of the 20th century.
So this may explain how this Arizal song custom filtered through many different Bentscher versions, even those not Chassidic, but only for the third meal.