Can we list the potential ways of transliterating Polish Ashkenazi בֵּית, בֵֿית, פֵֿא

[squee100]

Channalee, after looking at the sources I've provided which seems more reasonable?

Bys, Vys, Fy
Bies, Vies, Fie

Would Beis, Veis, Fei work for that sound?

The "ei" letter combination is often used in transliteration, as in the example of the Yiddish name "Sheina."  The late Rabbi Shmuel Gorr used this form of transliteration for his highly popular book Jewish Personal Names: Their Origin, Derivation and Diminutive Forms.  This book, by the way, is also the source-book that Chabad uses for their lists of "Popular Jewish Names" on their website.
 
YIVO, an internationally recognized Institute for Jewish Research, recommends using "ey" (rhymes with "grey," "prey" and "Sheyna") for Yiddish transliteration. I've linked their chart for you below.  Scroll down on the chart until you get to the "letter combinations" at the bottom.
 
https://lith520.class.uic.edu/Yiddish%20...0Chart.pdf
 
I suppose you can use whatever transliteration scheme that you choose to devise on your own but, if you want to be understood by the majority, I would suggest you stick to the YIVO standardization.  I personally don't care for some of YIVO's choices for transliteration but I understand that my preference is based on what I'm personally most familiar with (as I've already described in previous postings in this thread).

What about ei as in gesundheit (like you mentioned before)? Can you also please answer my question about bys/vys/fy vs. bies/vies/fie?