The dispute broke into plain view days ago when in support of Josh Mandel, the former Ohio state treasurer, for the Republican Senate race, the Club for Growth ran a television commercial showing the candidate Mr. Trump has endorsed, the “Hillbilly Elegy” author J.D. Vance, repeatedly denouncing Mr. Trump in 2016.

Mr. Trump’s response was brutish: He had an assistant send Mr. McIntosh a short text message telling him off in the most vulgar terms. The group, one of the few that actually spends heavily in primary races, responded by saying it would increase its spending on the ad.

That escalation drew an angry response from Donald Trump Jr., the former president’s eldest son, who had spent months urging his father to support Mr. Vance and has invested his own energy and influence on Mr. Vance’s behalf.