Have you considered that this pattern button: i.e. Star and Crescent, is really an Ottoman Army button?
The same pattern was adopted by the British Mandate for Egypt and Sudan until the monarchy was instituted in Egypt.
There are others made in Austria for the Ottoman Army and in Egypt and England for the Egyptian forces.
However, don't exclude that the Paris make could be an issue for one of the French spahis or zouvaves regiments of their North African armies. It would be interesting to know whether the Pellican Guard had just found this button available and adopted it! Unfortunately I'm not into US buttons.
Feel free to contact me.