When Representative-elect Sarah McBride, a Delaware Democrat, won her race for the House this month, becoming the first openly transgender person elected to Congress, she knew she would face attacks from hard-right Republicans over her identity. She just didn’t expect they would start before she had even been sworn in. In Washington this week for new member orientation, Ms. McBride was still sitting through mandatory cybersecurity trainings, setting up her payroll, selecting district offices and learning how to introduce a bill when her new Republican colleague, Representative Nancy Mace of South Carolina, announced plans to introduce a measure to bar...