MENOMONIE, Wis.—After the U.S. placed tariffs on European steel and aluminum in June, Europe hit back with a tax that, among other things, made American kidney beans 25% more expensive in Europe.

Now, Cindy Brown is running out of room to store the beans. One-ton bags of them cover the floors in her cavernous warehouses. Smaller sacks are piled on wood-pallet shelves. Kidney beans fill tall steel bins that dot the grounds.

Chippewa...