M.O.
I think that if you already have experience with pneumonia and your daughter needs breathing treatments, then you should watch her closely. Check your "list" of signs to watch for, every dr has them. I sincerely think that you should never feel stupid or like you're overreacting when it's your child's breathing!
You should also be able to call your drs service to speak with a physician. More often than not, they'll tell you to go get it checked out.
Personally, I'd consider switching to a dr who has evening and weekend hrs too. I don't think MOST kids have breathing problems so young. So for you, it's more important to have someone who's going to be available more than 9-5 M-F.
You might consider going to an urgent care center if needbe, instead of an ER. Urgent care should be able to treat and release her during the daytime hours. While the ER would only be when no one else is available or it's a crisis situation.
Since it's cold and flu season, you're likely in store for lots of colds/viruses and run ins with bacterial infections, like strep...most of which are highly contagious. It's just part of life.
As a Mommy, just try to remain calm and sensible. Watch your daughter's symptoms, check her temp and mind her breathing. What she has now may just be a cold. But colds can turn into earaches, sinus infections, etc. Just know what to watch for and when you need to get help.
Best wishes