Suppose that the United States government legally commissioned a massive Top Secret weapons project. The project picks up the pet name "Mickey Mouse" for reasons that would make sense if you were working close to it, and that name starts seeping into the documentation, code, presentations, and future naming schemes.
During the project, a small information leak reveals the existence of the name "Mickey Mouse" and corroborating evidence indicating that it's a military program that Disney absolutely doesn't want to be associated with.
Is Disney obligated/allowed to protect their trademark from US government or the contractor hired for the project, despite the fact that the details or existence of project Mickey Mouse can't be divulged?
As a side note, I'm choosing Mickey Mouse/Disney to abstract away the question of whether the trademark holder has the money or desire to pursue legal actions. I'm also taking it for granted that dumb pop culture references are constantly cropping up in R&D projects that would be incredibly stupid if a company lawyer happened to notice.