Two prominent civil rights agencies within the Trump administration are clashing over whether federal law protects lesbian, gay, bisexual, and transgender employees from discrimination. To avoid that legal uncertainty, some LGBT workers have a third avenue for bringing bias claims against their employers.

The catch? They have to work for a government contractor.

LGBT discrimination complaints more than tripled at a relatively obscure contractor watchdog office within the Labor Department over the last fiscal year, according to a Bloomberg Law analysis of publicly available government data.

“That data demonstrates that there is a need for vigorous enforcement of non-discrimination,”...