Check the "Castle" the US centennial exhibition, a red brick building pert of the Smithsonian on "the Mall." Natural History has some good displays about the original Americans. Also, the NRA Museum in nearby Fairfax VA. There are several Battlefields nearby. Gettysburg is only 120 miles NW. Antietam is even closer. Mannasas is 30 miles West of DC just a few miles from the NRA.
I had the very Good fortune to live in DC, when a truly great exhibit of Karl Bodmer's paintings and the artifacts gathered by Prince Maximillian were on display. A few years later, I was in Ft Clark North Dakota, where they had spent the winter about 1830, and the riverside and hills looked exactly the same as Bodmer had painted them. (Most of the watercolor portraits of Indians in books are Bodmer's pictures.)
Williamsburg and Yorktown are only 200 miles SE of DC. Ft McHenry in Baltimore is only 30 miles North.
As for fur trade, there really is very little on permanent display about that period in the Smithsonian's many buildings. Mt Vernon is 20 miles South, (I would suggest seeing Gunston Hall, George Mason's home, nearby but more impressive than Mt. Vernon)
Every summer there is an American Folklife festival emphasizing three or four American subcultures and it usually includes an Indian Tribal group. You would have to check if anything interesting is being featured
Looking for Fur trade era items around DC, is like asking for a Philly steak sandwich at a Chinese restaurant.