The underdog Israeli team that emerged as a sensation at the 2017 World Baseball Classic came with an asterisk. Despite the name of the country splashed across their uniforms, almost none of the players who helped upset powerhouses like South Korea, Chinese Taipei and Cuba were actually Israeli. Instead, the roster featured a collection of Americans participating under the tournament’s heritage rule, which granted eligibility to anybody with a Jewish grandparent or spouse.

Buoyed by that unexpected success, Israel now hopes...