The U.S. Patent and Trademark Office wants to preserve its power to refuse to register a scandalous or immoral trademark and wants the Supreme Court to take its side. But legal professionals say the office faces a steep climb over the First Amendment to win its case.

The PTO filed its petition to the Supreme Court this month against fashion designer Erik Brunetti’s bid to trademark his FUCT clothing brand—short for Friends yoU Can’t Trust. The office wants the high...