The question is, what is a Russian name? Most of tsars used Greek, Latin, Hebrew or Scandinavian names. Is Alexander a Russian name? Nicholas? Ivan (John)? Mikhail (Michael)? They only used Russified forms of those names.
IMO a lot depends on the religion. Should a non Russian-tsar be an Orthodox, he will keep his name in a Russified form (like John => Ivan). If not, he would have to convert to Orthodoxy and either accept a new name (like Catherine the Great) or keep using his previous name, again, in Russified form.
IMO a lot depends on the religion. Should a non Russian-tsar be an Orthodox, he will keep his name in a Russified form (like John => Ivan). If not, he would have to convert to Orthodoxy and either accept a new name (like Catherine the Great) or keep using his previous name, again, in Russified form.