I will shortly be writing my dissertation on the subject, and would like some debate on the matter. The main discussion I will be having is to whether there is a direct influence, or whether the American system simply became associated with the Roman Republic later.
I would argue that the association with the Roman Republic was produced later, partially due to a growing romantic view of the Early Republic era, and as a means of creating ties to the now heavily romanticized period. To me, it seems more like Mussolini using a fasces on a field of blue as his personal flag of office than anything. An attempt at legitimization, or to improve perception, through the creation of a false cultural history.