Director of - certainly he is the one that runs the Hotel Beltrano network.
Director at - he is one of the several (or more than one) directors that works at the place; that is, at the Hotel Beltrano Network.
But then, as it is a network (so in several places) we may consider "at" meaning that those directors work "with" the Hotel Beltrano Network. That is, simply another way to say that they "trabalham para (a tal rede) tal rede."
It also could be translated into "na" (tal rede) but still in the sense of "para", since there´s some sort of intersection of meanings here.