Actually, they may not have a say. The affiliate contracts as I understand them have some must run conditions for primetime. I'm not sure what those contractual rules are so they may not be weasel words. If they do have a choice to carry or not carry then they are accountable for the decision to carry it. In order to put them on the spot we would need to know the contract conditions covering their ability not to run this piece.
Local stations always have the choice as to whether or not they will air controversial programming so contacting them would be worthwhile.
However, contacting CBS is also beneficial, particularly if you mention that you will also be contacting their sponsors as well.