Are there any Smithsonian museums that have an area where kids can eat bagged lunches?

No, bagged lunches may not be eaten in any of the Smithsonian museums. See "Tips for Visiting the Smithsonian" at:

http://www.si.edu/Visit/Security

I'll add that you may see people doing it (or trying to do it) in the various cafeterias, but it is forbidden.

The entire Mall is just outside many of the Smithsonian museums and eating is certainly allowed there (keep fingers crosses for decent weather).

Best two places to eat outside in the area:

1) Enid Haupt Garden (between the Smithsonian castles and Independence Ave).

2) The fountain at the Sculpture Garden (between the National Gallery of Art west wing and the Museum of Natural History).

If it's raining, you'll have to save the bag lunches for another time and just eat in one of the museum cafes...

The National Portrait Gallery has a large interior courtyard with tables where visitors (and local office workers) may -- and do -- sit and enjoy a bagged lunch. Just don't sit at the set of tables that are directly outside the small museum cafe.

Those areas are really restaurants, so it would be like bringing a bagged lunch into a restaurant and expecting to eat it there.

Shouldn't be any problem except in very bad weather, as plenty of space outside.

We ate on picnic tables outside of the American History Museum on a beautiful 60 degree November day.