If a President chooses to abide by a non-treaty agreement, he can, on his executive authority, but it does not have the force of law and can be overturned by his successor.
That’s from the U.S. (Constitution) perspective. However, to the original question...what about international law? Is there something there that states dual citizen heads of state can not enter into treaties? In other words, international law says they have no authority to represent a single country on some issue?
"He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur"
Since the Senate is squarely in Barry's back pocket, it's not inconceivable that they would ratify a treaty he entered into.
So....back to the question of weather or not a dual citizen has that authority to begin with.