I think the problem here is the wording of enhancement bonuses presented in the Core Rulebook. Now, my copy is fairly old, so some of the wording may be presented differently in more recent versions. According to the Core Rulebook (p. 461), "Magic armor bonuses are enhancement bonuses, never rise above +5, and stack with regular armor bonuses (and with shield and magic shield enhancement bonuses)."
Now, I can see the argument of the OP on the surface level, but in my opinion, the argument ultimately falls flat when you read the description of Bracers of Armor (p. 504-505). "Bracers of armor and ordinary armor do not stack. If a creature receives a larger armor bonus from another source, the bracers of armor cease functioning and do not grant their armor bonus or their armor special abilities. If the bracers of armor grant a larger armor bonus, the other source of armor ceases functioning."
This can be interpreted in a few ways, but the ones relevant to this post would be as follows. Enhancement bonuses applied to armor modify the armor bonus of the armor in question, and would not be a direct enhancement of an individual's AC.
The other way would be that Bracers of Armor is an armor bonus and would stack with something labeled as an enhancement bonus; however, the last two sentences of the description for Bracers of Armor in essence states that the bonus for Bracers of Armor is mutually exclusive with any other source of an armor bonus.
Going back to the OP's example of a haramaki with a +5 enhancement bonus, the haramaki's armor bonus is only a +1. According to the second interpretation, Bracers of Armor +8 would render the haramaki's magic bonuses and abilities completely inert if both are being used by the same individual.