As someone who has one of these coughs, we know it annoys the people around us. It annoys us too! We would give anything for it to stop but for me, it doesn't between the months of October and May. It hurts to constantly cough. Abs are always sore, chest and throat is always raw and some times taste like iron/blood. It also makes it awkward to laugh when a co-worker says a joke and you have to choose between letting yourself laugh or risking a cough that can make you pass out (yes, I have passed out from laughing at work due to my cough).
I am sorry it annoys you and you have health concerns. Please also try to be considerate of the fact we are very conscious about it! When I first started working at my new job, the lady on the other side of my cubicle complained to my manager about my cough and he sent me home early a few times. It's rather embarrassing and frustrating because it is something I can't control.
To solve your issue, I would also look to take the more patience route. If you are allowed headphones/music try wearing it for part of the day. See if there is an empty desk you can move to that is further away.
Also talking to your own manager may not be a bad idea. Just because they aren't his manager doesn't mean they don't know about him. His manager may have emailed all the other managers to inform them of his cough should there be any concerns. Your manager might tell you that it's just an asthma reaction, or they may say, you know, I have noticed this cough too. I should find out if it has any contagious implications since some of you are concerned.
Please also note that PTO/Sick days are valuable. If I stayed home every day that my cough was bad, I would be out of vacation days within the first 2 months of getting them. If your company offers work from home and he is able to do so, maybe your company can offer him the ability to work from home on days he feels really bad.
Either way, Good luck! I hope I was able to at least give you a first hand account of what it is like from our side with the goal that you may end up being a little more understanding.