Indiana University gives this information:

Title: CHI-BABA CHI-BABA
Alternative Title [subtitle]: MY BAMBINO GO TO SLEEP
Lyricists/Composers: Mack David, Al Hoffman, Jerry Livingston
Published: New York, Oxford Music Corporation, 1947
First Line: Many years ago in old Sorento a certain ditty was quite....
First Line of Chorus: Chi-baba, chi-baba, chi-wawa, enjalawa, cookala goomba

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Now that we have some of the "authentic" spelling from the sheet music, I have to modify my views somewhat. Although all the transcriptions I found on the Internet were spelled phonetically, probably based on audio recordings, I now believe the original lyrics were also spelled phonetically, according to "English" spelling conventions. That is, the writers made no attempt to use Italian-style spelling, but instead wrote nonsense words that were designed to sound (sort of) Italian when read and pronounced by English-speakers.

For example, the traditional Italian alphabet doesn't have the letters J, K, W, X, or Y. An Italian writer would use the letter combination "gi" to represent the "j" sound, as in mangiare (to eat), Giovanni (John), Giuseppe (Joseph).

I'm not too surprised; the songwriters' names (David, Hoffman, Livingston) certainly don't sound Italian.

The question remains whether the writers were trying to imitate the sound of some authentic Italian words.

(I am a little out of my depth here, though; I wish some fluent Italian speaker would comment on this.)

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Besides the original 1947 sheet music, I found that CHI-BABA, CHI-BABA (MY BAMBINO GO TO SLEEP) is included in the following songbooks:

Hal Leonard Publishing Corp., "The 1940s: Piano, Vocal, Guitar", 2005.

Jeffrey Weiss, Martha Saxton, Rupert Holmes, "The Forties", Consolidated Music Publishers, 1975.

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Now, as to whether "goomba" or "goombah" is an insulting term— All I can say is that I grew up on the fringe of an Italian neighborhood in St. Louis, and I had several friends who were Italian ("Italian-American" we would say today, although we didn't use that term then; my "Italian" friends had parents or grandparents who were born in Italy), and I don't think I ever heard the word "goomba" in those days. (I was born in 1947 and left St. Louis in 1965.)

The only insulting term I ever heard was "dago," and that was pretty rare. (Then I was surprised when I moved to St. Paul and found "dago sandwiches" on the menu of some small local cafés, and people saw nothing wrong with it. But then, there are relatively few Italians in St. Paul.)

I do believe I heard the word "goomba" used on TV, in a friendly or bantering way, maybe by Dean Martin on his TV show, probably addressed to one of his Italian-American guests—but that was later, 1965 to '74.