If the aliens are required by their government not to draw unnecessary attention to themselves then one thing they cannot do is stand in an immensely public place for days on end without sleep or food.
In 2003 the American magician David Blaine spent 44 days suspended in a glass box over London, going without food for this time. It was a media sensation.
Admittedly, your alien ambassador isn't suspended in a box. But standing outside the White House for weeks would be noticed without the box, and even without the display of superhuman powers. The protester Brian Haw became famous for simply camping in a tent in Parliament Square for years. The whole point of doing this right outside the Houses of Parliament was to be noticed by the media. Protests take place outside the White House for the same reason.
The alien needs to make his vigil somewhere where the public won't notice him but the authorities will.
Presumably the aliens have the means to tell where, say, secret work on Black Projects is carried out - the equivalents of Area 51 that aren't famous. Since there really are extraterrestrials and UFOs in your universe, it's more than likely that the US government has some idea of this and a place to research the issue. If he knows of this place, the alien ambassador can probably spare himself the tedium of standing outside for ages. If he presents himself at the highly secret front door displaying detailed knowledge of what is done there he will be taken seriously. All he has to do to be believed completely by the base personnel is get them to take an X-ray, which given that this place actually is a base for research on aliens, should be easy to arrange. He won't get dissected. No one murders a peaceful emissary from a power which has already demonstrated superior technology for the trivial benefit of seeing how his insides work, particularly as he is offering to demonstrate that voluntarily.
None of this is remotely as cool as Wingman4l7's answer, of course. But I think taking a side in the domestic politics of primitive species is against the First Contact (Isolated Planet) Protocol & Code of Conduct.