Dear David,
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.
Tsvi ROZENBLAT
Herzliya, ISRAEL