"This is a public archive," Ms. Kahn-Leavitt said. "This should not be offered on an exclusive basis to anyone,..."
And this is what's at the heart of the matter. They have no business making such a deal with anyone. It should, by its very nature, be null and void for that reason alone!
Surely a simple answer for their financial woes would be to charge a fee for the cost of providing assistance to any film maker.
Another alternative would be that they get some share of any profits the production might make. The board would have to be totally out of touch not to realise the huge dent this is going to put in their reputation. I'm guessing that the Smithsonian relies heavily on bequests for much of its funding and the items in its collections, no doubt this will make people think twice about making such donations.