President Donald Trump’s new trade deal to replace NAFTA lets him boast that he achieved a campaign promise and claim vindication for his hardline negotiating tactics—just in time for the November midterm elections.

But he must convince a skeptical American public—and Congress—that a rework of the North American Free Trade Agreement will provide substantial economic gains for workers who bore the brunt of globalization over the past two decades.

His effort to promote the new accord may be critical for...