It would help to have a photo of the "government" sign.
www.flickr.com...
This is what I see around actual government bases. Well except for the bullet holes.
There seems to be no penalty for putting up a no trespassing signs where you don't really own the property, or you are misrepresenting yourself. You
find this in rural areas where Joe Hillbilly doesn't like people driving on "his" dirt road.
On BLM property, you might get them to remove such signs. They will absolutely remove chains or other illegal gates if you give them locations. Fake
signs seems to be less of a priority.