Other mistake: During the scenes in Beirut, we see a picture of a militiaman firing an RPG at an armored vehicle. If you look at the launcher, you can see that there is no grenade in the launcher when it is "fired."
The hat that Brad Pitt wears in the 1985 scenes is a Padre's hat with navy, orange and white as the colours. In 1985 the Padre's colors were brown and yellow. The color change didn't come until late 80's or early 90's.