Thank you very much for the clarification, Gerry.
I first assumed that it was a Klezmer song because Karen's video was of a Klezmer band and it was the tune only without words.

I couldn't find any recordings of the cigarette advertisement song, but searching for it led me to more recordings of the traditional words.

Here's a website that has several recordings of the song to the tune that interests us:
https://www.zemereshet.co.il/song.asp?id=3152&artist=1852

And here's a recording of the words to another (presumably the older) tune:
https://www.zemereshet.co.il/song.asp?id=10987

And a quick google translation to English:

According to Eliahu Hacohen, in a lecture entitled "The Book of My Childhood Poems," the source of the melody in an advertisement for "Salam Alikum" cigarettes in Germany by a Turkish band. A Russian presentation on the web gives the full story. Here is the translation of the subtitles by Uri Yaakobovitch: 1. "We brought Shalom Aleichem." Background of one song; 2. German orientalism; 3. Synagogues; 4. Industrial buildings; 5. "The Tobacco Mosque", Ynidze *; 6. Architecture and Nazism; 7. Turkish tobacco; 8. Photographs of three boxes of cigarettes "Salaam Aleikum"; 9. German advertisement for Salaam Aleikum; 10. German radio. All Germany hears the Fuhrer from the shelter; 11. We smoke "salam alikum" (3X). We smoke salaam, salaam, salam alikum; 12. Soviet dissidents, let's go south and north! Fear and terror for enemies! And that Golda Meir will lead us to the one-eyed war god Moshe Dayan; 13. Back to the commercial: Israir flight from Berlin to Israel.
[*] The "Tobacco Mosque" in Binzedze was an Oriental cigarette factory established in the Jewish Dresden by Hugo Zietz.

Eliahu Hacohen (born 1935) told the same lecture that learned the Hebrew song already in kindergarten. The words (in the first person only) with the appropriate rehearsals were found in Aya Ruppin's private notebook, where the songs around him were put in place around 1939. The melody was found in print in an American poem for Jewish soldiers in World War. The "Davar" newspaper documented the song "We brought Shalom Aleichem" in various contexts, the earliest ones: a demonstration against the Mandate authorities (24.12.1945), the absorption of illegal immigrants (June 10, 1946), a toddler's song to Tom (June 20, 1947) A new moshav in the Negev (24.3.1952), and for the first time in artistic performance - the Polytek Choir from Finland (31.3.1952).

Additional Performance:

    Ahuva Zadok
    The Ran Singing Group
    Eartha Kitt
    Sabra Disco (Gali Atari, Tzruya Lahav, Riki Manor and Nava Baruchin) (1976)
    Sexta (also a disco style)
    The "Am Yisrael" group (first song in Rosary)
    Dancers and musicians from the Jerusalem Academy of Music and Dance greet the Taglit Birthright Israel delegation at Ben Gurion Airport (June 2018, video)

See "We brought Shalom Aleichem" in another melody.