General Discussion

As someone before pointed out, all current references to the Department of Defense on contracts, official documents like directives, instructions, MOAs, MOUs, and Regulations must be "changed" and re-signed.
Legal Personnel records and orders - like hiring documents, training records, Veterans Affairs, retirement and and other official action documents need to be updated, or otherwise identified as still applicable despite the change of the department that issued the document or directive/order.
Also, changing the Department can open up all sorts of chances for messing around with internal regulations that define limits to uses of force or lethality.. a modern Department of War can be defined to have other strategic goals than a Department of Defense might have - including loosening the criteria for "First Strike".
The reason the Department of Defense was created was to consolidate the various War Departments for the Navy and the Army (and Air Force) under a single funding, supply, and strategic umbrella that Congress could manage at all times, both in peace and war, after the interdepartmental bickering and political issues that caused all sorts of waste and fatal miscalculations encountered during WWII (like - why were there no heavy cruisers at D-Day?)

It's not just changing the letterhead that's going to cost us with this change.