As if TV carriage negotiations haven’t become contentious enough, a new wrinkle is adding to the complexity: the rise of skinny bundles and “cord-shaving.”

In a brewing spat, AMC Networks Inc. is demanding that the National Cable Television Cooperative, a consortium of small cable providers, pay on behalf of all its subscribers—whether or not they actually receive AMC and its sister channels in their cable packages. If the two sides don’t reach an agreement by the end of the year, those channels could go dark on the systems...