The White House is increasingly at odds with congressional Republicans over the party’s health-care message for the 2020 election campaign.

A growing number of Republicans in Congress want to focus on transparency, competition and other free-market solutions to rising costs, all as part of their platform for 2020. But Trump administration officials are taking steps to invalidate the Affordable Care Act, including a move this week to help scuttle the law through the courts.

Shaping...