When Donald Trump’s children and campaign staff arrived in South Florida during the 2016 presidential campaign, they were often greeted at the airport by Zoltan Tamas in a black Cadillac Escalade. As a senior security guard at Trump National Golf Club in the town of Jupiter, Mr. Tamas was licensed to carry a gun. He bought a home, paid taxes and never ran afoul of the law since immigrating legally to the United States from Romania in 2011.

But for eight months, Mr. Tamas, 38, has been locked in a correctional facility six hours’ drive from his family as he fights a protracted legal battle to remain in the United States. During that time, he has not once seen his wife, 11-year-old son and 8-year-old daughter, who suffers from congenital heart disease.

Mr. Tamas has fallen prey to the crackdown on immigration that is at the top of the president’s national agenda — and letters of support from high-profile businesspeople and his former bosses have so far not helped him win leniency.

Mr. Tamas, a green-card holder, was arrested by immigration authorities after he applied for United States citizenship in 2016 and a background check revealed that he had been convicted in absentia of committing insurance fraud in Romania.