The only exhibits in the Smithsonian that do not permit Photography are usually the ones on loan. These are usually in the Galleries, and the rooms that do not allow photography have a sign up.
Most of the Museums allow unlimited Photography, but do not allow tripods or bipods. Most will allow Monopods, but best to ask.
I'll upload some more shots, it was a fantastic exhibit. It captivated Nikki, who saw it two weeks ago and wanted to go back so I could see it.