Once again, I am surprised at the focus on the coughing issue. This may be the more annoying of this man’s issues, but by itself it is, at least in my opinion, not a justification for asking him to leave. Asking him to stop, perhaps, but to castigate somebody for such a personal habit, even if it does disturb others, is unfair.
But the shanyim mikra, in my eyes, is more of an issue. He could do it another time (bein gavra is hardly enough time to cover more than half), he could be quieter, especially considering the character of the shul. The fact that he has been asked to stop and hasn’t indicates he isn’t the easiest person to deal with, and since this practice disturbs davening (in what I assume is a smaller shul), the situation should be dealt with, if possible with the participation of the Rav or Gabbe, as delicately as possible.
Although the tone in which the question is placed, and I apologize if I am being presumptuous, indicates that there is some sort of personality clash involved, that he may not be the most pleasant of individuals, perhaps leading to his difficult habits being exacerbated in your eyes. This is amueter psychology at it’s worst, and I am truly sorry if I am barking up the wrong tree, but am I in any way correct in my assumption?