I am also a government contractor and have been staying at Marriotts less and less in the past 2 years. I have had luck in the past where they either didn't ask for a government ID or were OK with me showing them a copy of my company's contract with the government. Lately the people who book travel at my work would rather me stay at Hiltons where rooms are usually within per diem without trying to game the system.
I think Marriott is losing a big chunk of business travelers because of this policy. I can see exempting prime locations, like Hawaii, NYC, etc. but I can assure the Marriott Corporation that I am not traveling to Dayton, OH regularly for pleasure.