I'm beginning to feel like you haven't seen all the episodes from the beginning till the end. There's nothing here that wouldn't make sense in the comedically nationalistic way it's supposed to make sense. What wouldn't, considering the incredibly simple way this show is proceeding?
Sending the troops to fight the dragon isn't a military matter until a permission has been granted. Just like the American military can't attack terrorist camps in some foreign country unless the politicians allow it (in a broad or specific manner, I don't know how they decide such stuff). When it's a political matter, things are muddy. Itami already has the attention of the political machinery because he was a hero when the fantasy world troops first attacked Tokyo, and he again did stuff in the other world by rescuing the refugees. Afterwards he was there when they saved the city, and again brought the imperial princess to Tokyo, which launched the peace talks. Even the opposition party asked him why he couldn't save more of the refugees. So, if he was there organizing the dragon removal mission, things could go smoother with the political deciders.