Going by the name, PulmaCort would be a cortico-steroid type med(NOT what some weightlifters abuse--whole different type of med). These help to reduce the mucous production in the bronchial tubes and operate by a different mechanism than the Albuterol for emergencies.
Albuterol and relatives work on relaxing the muscle spasm in the bronchial tubes, thus widening them, and having no effect on mucous at all. Well except that it can be coughed out more easily if the tubes aren't consticting.
The PulmoCort won't have the heartrate increasing side effects that Albuteral and such can sometimes have, so don't worry about one on top of the other.
I take both Types of meds and they work well together.
Also keep in mind that inhaled meds go right into the lungs, thus the doses can be lower than what would be needed with an oral med that goes everywhere in the circulation.
Of course the chewable may be easier to deal with. I have no experience with that since I'm the asthmatic, not the children.