Be prepared for disappointment. Unless things have changed since I went to the Smithsonian last June, the model isn't there.
The Smithsonian's ship models are (or used to be) exhibited in two separate sections within the Museum of American History. The "Hall of Maritime Enterprise," on the ground floor, covers civilian maritime pursuits - merchant shipping, whaling, fishing, riverboats, ocean liners, clipper ships, etc. That gallery was installed in the late seventies, and hasn't been changed much since. It contains quite a few ship models, most of them dating from the mid-twentieth century - part (though by no means all) of the collection built up by the late, great Howard I. Chapelle when he was curator there. As I understand it the gallery is due for an extensive renovation in the near future. (I suppose it's possible that it's closed now; it would be a good idea to check before going.)
On the top floor of the same building there used to be a gallery devoted to the history of the American armed forces. That's where the models of warships - including the Constitution, and the spectacular 1/48 Missouri - were displayed. That gallery was put up in the fifties and sixties. Originally it was supposed to tell the story of the American military from the beginning to the present day, but before it was finished the institution's interest started to wane. (Chapelle passed away, and during the Vietnam War military history wasn't the most popular subject among the public - or the Smithsonian's benefactors.) By the nineties the exhibition was looking pretty tired. There was, in fact, quite a bit of criticism to the effect that the Smithsonian wasn't doing justice to the armed forces.
That criticism was one of several factors that, just a few years ago, led to the removal of the old armed forces exhibition and the installation of a new, blockbuster gallery called "The Price of Freedom: The History of America's Wars." It opened last spring, and I got to see it in June. In general my opinion of it is extremely high. It contains some spine-tingling artifacts (George Washington's and Andrew Jackson's uniforms, Phil Sheridan's horse, a bullet-perforated tree trunk from the Battle of Chickamauga, a WWII Jeep suspended from the ceiling, a Huey helicopter that saw action in Vietnam, a North Vietnamese prison guard's uniform, etc., etc.). And the exhibition techniques are state-of-the-art. I've got a minor quibble about the treatment of the Jefferson administration. One exhibit case is devoted to the Lewis and Clark expedition (semi-military, it seems to me), but two events that I think are pretty important got left out: the Barbary Wars (first time the military forces of the newly created U.S. went into action overseas) and the founding of West Point. But in general I think the gallery covers American military history about as thoroughly and objectively as it could be covered in that amount of space.
Bad news: almost all of the old warship models are gone. I remember seeing Alfred Brownell's Rattlesnake, the Maine, and maybe a Civil War ironclad. That's about it. It seems that ship models just aren't "in" among the current generation of exhibit designers.
I don't know where all those models (there were dozens in the old gallery) went. When the Smithsonian takes artifacts off exhibition it generally makes them available to other institutions; it's likely that some, at least, of those models can be seen in other museums in the Washington area (or elsewhere). I only got a few days there last spring, so I didn't get around much.
Two other Washington-area sites belong on the itinerary of any ship model enthusiast. One - the Navy Museum in the Washington Navy Yard. It's a big, well-run place with a good ship model collection. (As a matter of fact I think I recall reading recently that the big Missouri ended up there - but I may be mistaken about that.) Definitely worth at least a couple of hours. Two - though it requires a side trip of about 35 miles, any ship model buff needs to beg spousal/family permission for a pilgimage to the U.S. Naval Academy Museum at Annapolis. When you get there you need to make two stops in the same building: the museum itself on the main floor of Preble Hall, and the (relatively) new Rogers Gallery in the basement. The latter contains the most spectacular collection of seventeenth- and eighteenth-century English "Board Room" models outside Britain, now exhibited in a gallery that does them justice. An absolute "must see." And right next door is the Naval Institute Bookstore, where you can run up a credit card debt that you won't be able to pay off in this lifetime.
But don't go to the Smithsonian expecting to see a distinguished collection of ship models. On second thought - better see the merchant ships in the "Hall of Maritime Enterprise" while they're still there. Something tells me they may not survive the upcoming renovation.