I don't know, I am just guessing because not a single American company has publicly stated they are against its product use up to a certain degree by DoD? If they have not publicly declared it then yes, DoD simply don't sign the contract.
(I think it's also the case that Anthropic didn't make any public statements about the negotiations until after the DoD was already publicly threatening this designation. But that seems less important than the question of: is it good for the US government to try to destroy companies whose public statements it dislikes?)