It's now the Department of War and war isn't known for its concern about looking good.
We all know how this will end, they know it too - both sides - ergo, it's a clear case of blame washing - Anthropic will do everything they're told but will keep a smiley face and the image of a "fighter for the people". DOW will absorb the blame like a sponge and will ask for more, not necessarily from Anthropic.
(By the National Security Act of 1947 and its 1949 amendment, it is the Department of Defense, and nothing short of an act of congress changes that. The executive order secondarily naming it the department of war has as much legal weight as my personal order naming it the Department of Brainrot or Whatever.)
The administration proudly call it that, formal documents call it that, and war they wage. It is a Department of War.
A remarkable exception of non-doublespeak nowadays.
Point stands though that the 'pieces of paper' are special, even if the people handling them are lawless. In the long term, cosplaying it as Department of War is just that.