Donald J. Trump knew that JD Vance could take a punch. But during their first week together on the campaign trail, the former president wondered just how many hits his new running mate could absorb.

The volume and velocity of attacks from Democrats stunned even Mr. Trump. He was unaware of the most incendiary remarks that opponents were rapidly unearthing from Mr. Vance’s past, and the former president told allies that he was troubled by the idea that more comments would come to light as Democrats savaged his heir apparent as weird and anti-women.