Spycraft: The Great Game. Thanks for reminding me that game existed. It was locked in foggy memories I had from watching a guy stream it on twitch way back, and I only halfway remembered the game was even real, but your description brought it all back. From a quick search this intel report in particular stands out to me and confirms that it's the one I'm thinking of.

(On a weird personal note: I now recall while watching that stream on one monitor I was playing Nier Replicant on another, so the memories of both games got jumbled together into one, making it even harder to pin down.)