DNA on the sly would be worthless. You could never use it as evidence.
You would know the truth and better know then which direction to investigate to come up with the facts. You would know whether it would be worth going to the trouble of possibly getting it legally. You really don't know where the knowledge can take you until you find out what the truth about his genetic background is. Plus you could publish it and let Obama have to deal with the fact that, though legally not usable, the truth is out in the public. You could then pressure him to give his DNA to try disprove what you have found, telling the world that he can't.