WASHINGTON—Senate Minority Leader Mitch McConnell recently delivered a direct pitch to one of America's top defense contractors: Get off the sidelines and start backing Republicans who will protect military spending.

Mr. McConnell's message, according to people who attended a Nov. 6 fundraiser with officials from BAE Systems Inc., was that if Republican lawmakers lose primary elections from challengers further to their right who have few qualms about trimming military budgets, the defense industry could expect dwindling support...