Oklahoma’s conservative Republican governor, Kevin Stitt, and its conservative Republican schools superintendent have appeared to stand shoulder-to-shoulder in the years of Donald J. Trump, with the former sending Oklahoma guardsmen to the southern border and the latter stocking the state’s schools with Trump-branded Bibles.

But when the superintendent, Ryan Walters, proposed finding the undocumented students in Oklahoma’s schools, Mr. Stitt said enough is enough.

“When I saw them…