The bathrooms off Peacock Alley in the Willard Hotel were just renovated. Lovely new wallpaper. Hotel Monaco also has some great wallpaper and fixtures. Mandarin Oriental/now the Salamander is pretty with very nice soap and hand lotion (assuming the new owners didn't change much). The second floor bathroom at American Art/Portrait Gallery that is by the Luce Center is a family bathroom so you have a big room to yourself.

The FDR Memorial has the best bathrooms on the Tidal Basin. Very utilitarian but always reliable plus they have the heat on in the very early morning when I'm down there to take dawn photos during cherry blossom time.

They will be crowded but not summer tourist season crowded. You might want to avoid or quickly go through choke points like the area around the Hope Diamond at Natural History. The Objects of Wonder exhibit is usually not crowded even though it is a wonderful mix of items because so many people are in the Gem & Mineral Hall. Same with the others - the museums are big enough you can find areas where you will be more comfortable. American Indian has a gorgeous exhibit of Preston Singletary glass that you shouldn't miss.