
[@Dr_Igor ] According to YIVO transliteration standard https://www.yiddishwit.com/transliteration.html, it should be "Bay" and "Sheyn"
Also, if you write in the same place "bei" and "shein" - why would you read "ei" as "ey" in one word and as "ay" in another?
Net, I'd recommend modifying the title in the other variants if the link is needed.

Bei and Shein are dominant everywhere. We can't change the world because YIVO thinks we should. They can G&FT. The irony is that even in the text of the song on this page it's Bei and Shein. Look at the list of all "also performed by" here - a minority successfully link - 3 out of 16.

Igor, I'm not sure I get your argument. Yes, the Google search gives only 23k hits for "Bay Mir Bistu Sheyn" and 30k for "Bei Mir Bistu Shein". I would not call it "dominant" but even if so, what does it prove? That there are not so many people in this world anymore that are literate in Yiddish? But we know it already...
And there are also versions that are spelled "Bei Mir Bistu Shoen", "Bai Mir Bistu Shein", "Baj Mir Bistu Shejn" and so on. What do you plan to do with them?

My argument is very simple - the transliteration should produce the "transliterated word" so that if English speakers pronounce it it would sound the closest to how it sounds in the rendition. No English speaker would pronounce Bay as it sounds in this or any other rendition. Why English? because it is the universal language on this planet. The transliteration should give regular people across the globe and on this site the best idea of how it sounds. The number of Google hits is irrelevant - we should go by how the majority of singers pronounce it.
https://en.wikipedia.org/wiki/Bei_Mir_Bistu_Shein

It is interesting, that the very link you provided, an English-language wiki page, offers several spelling options:
"Bei Mir Bistu Shein" (Yiddish: בײַ מיר ביסטו שעהן, or Yiddish: בײַ מיר ביסטו שיין, [ˌbaj ˈmir ˌbistu ˈʃejn]) ... transcribed the Yiddish title as "Bay Mir Bistu Sheyn"... and ... under a Germanized spelling of the title, "Bei mir bist du schön"

Also checked the French version of the song - they spelled German-like like Andrew Sisters originally did: "Bei mir bist du schön"
https://lyricstranslate.com/en/léo-marjane-bei-mir-bist-du-schön-lyrics....

Let me say it again. I don't (and we should not) care how some web site owners, youtube posters spell it out. They all (try to) provide a trasliteration of how Yiddish sounds. Those who choose "Bay" either don't speak English or just copied that spelling from somebody who does not. And this is the Andrews sisters' page, right? https://www.amazon.com/Bei-Mir-Bist-Du-Schon/dp/B00005LQES
I care much less about schein (though schon does not make sense, schön - maybe because it is clear English pronunciation should not be used) or if it's bistu or bist du, but it can't be "Bay".

There are some and more to be added
https://lyricstranslate.com/en/melissa-kross-bei-mir-bistu-shein-lyrics и у Gunhild Carling подредактировал название чтобы залинковалась
с Мелиссой. Многие из тех кто на сестрах указаны на их страницах песни нет. Будем добавлять.
Расхождения в bistu/bist du , shein по-прежнему будут препятствовать прямой линковке.
Lyrics: Jacob Jacobs / Sammy Cahn
Composer: Sholom Secunda
https://en.wikipedia.org/wiki/Bei_Mir_Bistu_Shein