I would actually hesitate to call The Little Prince political satire. Rather than being interested in ideas or instances related to government or power, Saint-Exupery is actually considered a humanist. He is more concerned with social issues, issues having to do with society and the heart, rather than politics. The only chapter we can really see as having political connotations is Chapter 10 in which the prince visits the king who is really reigning over no one in particular.
Literary critics have recognized this description of the king as fitting a lot of authority figures, but especially of the French government before and during German occupation early on in World War II (eNotes, "Social Concerns/Themes"). France's initial response to Nazi Germany was to join forces with Great Britain in trying to appease the Nazis. France signed the Munich Pact along with Great Britain and Italy, allowing Germany to invade Czechoslovakia. When appeasing the Nazis failed, France went to war with Germany but surrendered as soon as 1940. The Nazis set up a new French government in Vichy, headed by acclaimed World War I French general Henri Philippe Petain. Under Philippe's direction, the French "acquiesced in the plunder of French resources" and sent French to labor camps in Nazi Germany, all because they "hoped to preserve at least some small amount of French sovereignty" ("France History--France during World War II"). Since Saint-Exupery was a French pilot who had been sent to the US to try and convince our government to take action in the war, the actions of his country would have been very important to him.
We see the actions of his country reflected in the king's insistence on pacifying his subjects. When the prince yawns before the king, the king forbids him to yawn further, saying that it is "contrary to etiquette to yawn in the presence of a king" (Ch. 10). When the prince argues that he can't help yawning because he is tired from his travels, the king changes his decree and orders him to yawn. When the prince states that he is now too frightened to yawn, the king sputters the ridiculous decree, "Then I--order you sometimes to yawn and sometimes to--" (Ch. 10). Both the king's inability to pass sound laws plus consistently uphold any ridiculous laws he makes with the intention of pacifying his subjects, portrays the king as a weak governor, as weak as the French government at the time of World War II. Like the king in The Little Prince, the French government also acted with the sole intent of pacifying Germany, creating a very ironic French government. However, since the rest of the book is about social issues and exposing human nature, it does not seem that Saint-Exupery had political motives for writing this chapter. Instead, it can be said that Saint-Exupery had the intent of exposing human nature as ridiculous and unable to see what's truly important, just as we see throughout the book.