You wouldn't need an identical controller, just one that has an identical line protocol. In this case it's a USB HID Device that prescribes to the XInput Hardware API. Logitech, Razer, and others already make compatible devices.

There have been third-party Xbox controllers almost since the console came out. No doubt ONE of them will gobble up a DoD contract to continue production if it's offered. (Former Submarine FT)

The military has been buying 360 controllers to do a bunch of different things and I'm sure MS has been providing them on contract and will continue to provide them because it's easy money.

Yes

Xinput is a pervasive standard at this point, to where there is a large market of third party controllers and software support in various linux distros as well as a JS api to access it in Chrome and Firefox.

As long as the military doesn't buy them on Amazon.com, yea they probably would. Escrow of technical and IP documents is pretty common in industry procurement, I'd imagine that it's even more common in procurement for defense.

Given it's military, they're very likely already sharing NOW.

Dude, it's Microsoft. Most of their business is to the government.