SHANGHAI—A son of the U.S. ambassador to Beijing used his connections to President Donald Trump this week to drum up business for his public-relations firm.

Eric Branstad, a son of Ambassador Terry Branstad, and the U.S. Commerce Department’s liaison to the White House until January, spoke on Thursday in Shanghai to more than 100 lawyers, bankers and advisers at a seminar titled “How to React To (Potential) US-China Trade War?”

Mr....