Political conflicts in the Middle East between the Saudi-led camp of Sunni powers and a rival Shiite camp led by Iran have already morphed into a religious war. Now, a theological dispute within Sunni Islam is causing another regional political rift—a result of an initially obscure conference in Russia’s Chechen Republic.

Chechnya’s strongman Ramzan Kadyrov—just re-elected with a modest 98% of the vote—is a follower of the Sufi current of Sunni Islam. The diverse and more mystical version of the Muslim faith is one long at...