The "melech" of the free world ordering a burger as he becomes "melech".
Irrelevant. Shulchan melachim does not mean what a melech himself eats but rather what he serves at an official dinner.
This is an old issue when certain people tried to make trouble about eating sardines because there is a brand of sardines named King Oscar sardines. They argued that if the King of Sweden ate sardines then it proves that it needs to be bishul yisroel. S"A clearly speaks about the "little fish" that are not choshuv.
Anyway, Obama eating burgers is the same as that.