The Hebrew inscription on this headstone is of the given name (Eliyahu) and name of father (Tsvi), without the surname.
In many cases Jews in the diaspora adopt or given a non-Jewish first name, while keeping their Jewish first name (sometimes even in a non-formal way).
My guess is that the name ALEXANDER is the non-Jewish name chosen due to some phonetic resemblance to Eli (which is the short form of Eliyahu).
Q: "What name would Alexander use in any records found in the Pale of Settlement"?
A: In case that ALEXANDER is a name added upon immigration, I would look for Eliyahu אליהו - Most likely = ILLIYA /ELLIYA or ELI (His father: Tsvi צבי = TSVI/ ZVI/ HIRSH/ HERSH/ GERSH/ /GERSHON/ GRISHA/ GREGORI)
Good luck with your search.