Wouldn't even really be a fight. Wouldn't even really need to fight much.
Rural America has access to and controls all the infrastructure that makes a modern economy possible. Just cut off or destroy the highway and railroad bridges/tunnels, transmission lines, and pipelines flowing into the major coastal cities and wait for them to starve and freeze in darkness until they quit quickly enough.
Both sides definitely lose in any material or moral sense, sure, but in a strictly military sense... I'm betting on whichever side controls the mountain passes through the Appalachians into the major population centers in the Northeast, and I think we both know which side of any conflict like this that would be.