You might consider Jamestown Settlement.
They do a pretty good job on the history and culture of the Powhatan Indians. I’m not sure if the cafe at the musuem sells any Native American fare or not. I don’t think so. My husband (works in costume in the recreated English Fort area) has complained about this – they’re missing an opportunity. Let me know if you’d coming to town and we can recommend other places (more Roadfood than native).
The Museum of the American Indian in DC is wonderful. Look at their site and get the times for the guided tour. It really helps you understand a lot of what is going on in that museum. They have a wonderful, albeit pricey, cafeteria severing upscale native and native inspired recipes.