R' Menachem Meinish Chayes of Brody&Florence - Chajes/Chayut

R' Menachem Meinish Chayes of Brody&Florence is my wife's 12th great grandfather.
I think it should stay Chajes/Chayut as the first spelling is the pronunciation in ashkenazi hebrew and the second is the transliteration of modern Israeli pronunciation

Since Geni has a place for the modern Hebrew language / orthography version of his name, that's where it should go - and it's there already.

The rule of thumb that most of us who are trained in history and linguistics follow is to have the name match the documented evidence and our knowledge of the what the name would have been written / pronounced historically. To the best of our ability. My background in PhD in linguistics, training in history undergraduate and afterward, and background in the linguistics and history of Semitic languages, including Arabic and Hebrew, and in Yiddish linguistics.

I agree with Vivian and Michael and point Shimon to the Hebrew version which matches his desire for there to be a modern Israeli pronunciation.

My two cents.

R' Menachem Meinish Chayes of Brody&Florence is only my third cousin 11 times removed

He is right