A fight between America’s two biggest brewers is jeopardizing a proposed “Got Milk?”-style campaign intended to help struggling beer makers win back drinkers who have defected to wine and spirits.

Anheuser-Busch InBev SA, Molson Coors Brewing Co., Heineken NV and Constellation Brands Inc. have for over a year discussed a potential multimillion-dollar, brand-agnostic campaign aimed at improving the overall health of the beer category.

But...