As a thought for who to approach about it, do you know if the George Eastman House has any of your father's work? They're certainly the pre-eminent photography museum here in the US. If not them, perhaps the Getty out in Los Angeles. Also, out of curiosity, when you say "The Smithsonian", which museum in particular are you speaking of? The Smithsonian is a very large institution with many museums. There is the National Gallery of Art (which is actually NOT part of the Smithsonian), the Museum of American Art/National Portrait Gallery (probably the best museum in DC for exhibiting photography at this time), the Hirshorn Museum (modern art, with a fair photography collection) and the Renwick (American arts and crafts - dunno about their photography collection if any, but they might be worth a shot). Non-Smithsonian collections here in Washington DC worth approaching would include the Corcoran Gallery of Art and the Phillips Collection. In New York, the obvious choices would be the Metropolitan Museum of Art, MOMA, and ICP. Also think about universities - RISD (Rhode Island School of Design), Maryland Institute College of Art, etc.
Given that your father's Hillotype is not in fact an "original" Hillotype but a reconstruction of the process, perhaps a science museum would be a better choice to approach. They might have greater interest in it as a material object, and would be willing to pay (perhaps even pay more) for something that an art museum might pass on.