Director Adrian Lyne explains the timeline in a commentary
In a commentary by Adrian Lyne, which you can listen to here, there are two sections which I recently remember listening to which I think pretty much confirm that it was his intention that the events after Vietnam are in Jacob's imagination.
In the very beginning, around the 1:14 point, Lyne mentions that when he first read the script of Jacob's Ladder it reminded him of a short film by Robert Enrico called "The Incident at Owl Creek" which was actually shown on the The Twilight Zone Season 5 Episode 22 "An Occurrence at Owl Creek Bridge" and is based on a short story by Ambrose Bierce.
The film is about a man who escapes execution at the last second at the beginning of the film only for us to see at the end of the film that he didn't actually escape and everything we've seen was in his head in split seconds before the rope kills him. He mentions that this short film was the basis he had for Jacob's Ladder.
Later on in the film, in fact very close to the scene you brought up, he goes into a deeper explanation of the film's timeline (this is around the 1:38:18 mark):
Its a complicated movie in that it deals with different time sequences. It deals with Vietnam that you think is in the past and in fact isn't, in fact Vietnam is the present. You are dealing with the sequence after Vietnam when he lives with Jesse and you think that this is a reality when in fact its just imagined while he's dying in Vietnam. And you also have to deal with his memory of his relationship with his wife. So, its a very complicated movie...
So is the pin a goof?
I think its anyone's guess but after watching the scene again with this in mind I have a personal interpretation that still fits with the overall timeline theory of the movie. I don't feel entirely guilty adding my own personal interpretation here, given this movie is so brilliantly and artistically made that it has led to so much personal interpretation by its viewers. In fact the video you linked to says something along those lines, how there are levels of symbolism that exist beyond the director's original intent, and this is what makes the movie great art.
The scene - When he is in the cab he notices the driver's license with the 1972 date and he starts to grab his side in pain. At this point he flashes back to Vietnam where he is being stabbed with a bayonet in the same spot that he is groping in pain in the cab. As the film bounces back and forth between these scenes we see the Nixon Now campaign pin on the dash, but it is blurred out.
So what is happening? My theory is that he starts to notice little elements that are out of place in reality (aka Vietnam 1971). These elements trigger his existential suspicions that something is off with the 1972 reality.
Seeing the campaign pin triggers him to come out of his imagined reality in 1972 and back to the real one in 1971 Vietnam.