A rabbi in St. Louis Park, Minn., was more than six thousand miles from Jerusalem when he heard the Israeli government decided to bar two Muslim members of Congress from making an official visit to the Jewish state.

But within minutes, his phone was flooded with calls from congregants, local Jewish agencies and lay leaders who plunged into what had become a familiar routine: Figuring out how to respond to yet another political battle over their congresswoman, Representative Ilhan Omar, and Israel.

“There was very much an attitude of, ‘oh, here we go again,’” said Rabbi Avi S. Olitzky. “The pendulum keeps swinging left and right, left and right. It’s dizzying and exhausting and distracting. Emotions are raw.”

For months, American Jews in Ms. Omar’s district and beyond have found themselves enmeshed in a deeply uncomfortable debate over the growing distance between traditional liberal American Jewish values and the political realities of an Israeli government that’s embraced hard-line policies and a deep alliance with President Donald Trump. On Thursday, in one of Mr. Trump’s most audacious moves yet, he successfully urged Israel to deny entrance to Ms. Omar and Representative Rashida Tlaib, who planned to tour the West Bank.