OMAHA, Neb.—At a town hall meeting on Saturday, Rep. Don Bacon (R., Neb.) tried to tread a middle path on race relations in America, on the ban of transgender people joining the military, on the pardon of former Arizona Sheriff Joe Arpaio and on health care, among other topics.

It was a tough sell to an audience that included both Democrats opposed to President Donald Trump’s recent actions and Republicans delighted by them, according to interviews with several who attended and their responses to his remarks.

“We’ve...