If the cough is totally disruptive, makes them possibly gasp for breath (I have one of these coughs right now and just can not shake it...I can not imagine a child dealing with it), I could see an exclusion. I run a home based childcare and certainly would consider it. They get to make the rules, remember that. I see many people posting things like "you pay them, they can't keep your child out" sort of mentality. Well...yup, they can. Most policy books have a "we get to make the call on an individual basis" sort of line in there somehwere. I know mine does. I am the boss here, and its why I get paid the big bucks (LMAO..ok...that was funny..sorry!)
Anyhow, you are not their boss, you are a client, and truthfully, if you are not happy with their policy on this (once you have exhausted the discussions and "what can we do" sort of stuff of course), you are free to move along elsewhere to another care facility.
As far as the cough itself, a trip to the doc perhaps, altho, personally I hold little stock in their replies with such non-specific illness as this sounds. FOr my own, I finally went in since an old inhaler I had was expired and I knew that would help me (I get this at the tail end of many colds in the last few years).
Good luck!