Hmmmm,...what breed(s) of dog? Just out of curiosity.
"Very bad cough" could be anything from canine cough/upper respiratory, to serious heart problem, to heartworm infestation. Absolutely you did the right thing by knowing when to back off.
Sometimes I suspect they "pull the worst one off the shelf" for try-outs anyway, so one more GOOD reason to put the dog first.
Maybe they are critiquing potential hirees on more than a perfect haircut. In a perfect world that is, lol!
If you have dogs at home...I hope you shed and laundered all your clothing...just to be paranoid, errr...I mean safe!
Often it's not what you say, but how you say it.