As someone with misophonia this sounds like hell for your father and not like a great situation for you or the rest of the family either. The coughing is probably not a retaliation but rather sounds like a coping mechanism called mirroring which makes one feel a little more in control of what otherwise feels like like being mentally violated. The violence in case it's misophonia is not personal either, its just a fight or flight response to what can only be described as torture(it's never justified to be violent and agressive in the manner you have described but it very much cannot be helped if he has misophonia and is forced to be exposed to the triggers). Dont expect logic to get through as this illness is in no way rational and there is no throght process behind the outbursts. Exposure usually makes things worse and there is no therapy shown to really help misophonia. The most effective coping mechanism when being triggered is leaving the situation but that seems impossible in your situation. What i would suggest is buying him a pair of proper sound isolating headphones and pair them with some sort of earplugs. This sounds like a very frustrating situation for the whole family and it doesnt help that he is refusing to admit to the issue. Maybe you could print out some information about misophonia and give that to him instead of confronting him verbally.

If you have any questions about misophonia you can message me