At her age, why don't they have her on nebs when it is acting up? I'd seek another opinion, a pediatric pulmonologist if you can get into one. You obviously know it as well, but that many incidents in such a short time, means it is not controlled.
DD2 has viral induced asthma, she had struggled with RAD since she was a baby, she was 1 when she was first put on an inhaler. No symptoms other when she was ill which then meant she got sick very easily and very quickly. We did the same thing you are doing, winter after winter, it got to point where pneumonia was occurring several times a year, she was always ill, always coughing, and the coughs would linger and linger. We've had a standing order for codeine cough syrup for years now.
Her regular docs weren't concerned, "eh, it will get better when she gets older". This fall, I had enough, she was 4 by then and I wasn't about to go through another winter with multiple ER trips. By then she was on nebs and two different inhalers. Went to the specialists who couldn't believe how poorly her asthma was controlled and was started an aggressive treatment plan. Switched neb meds, we do those at the first sign of any illness, switched inhaler meds and instead of a pedi spacer went to a inspirease bag. And I finally agreed to give Singular a try. I wasn't thrilled about it but thought we could do a trial and things couldn't get worse then they had been. I have a entirely different child today. We've gone though some nasty bugs this winter, and NOTHING from her all winter, no coughing, no ER. DS got croup, she barely got a cold, last summer she ended in the hospital on a beach vacation from croup. DS and I again got this nasty, nasty respiratory infection that knocked both of us out for two weeks even after having to get antibiotics, she never got it. This child previously was nicknamed the walking petri dish because she got every germ known to man, if a disease came through the state, she got it, and got it severely. I can't sing from the mountain tops loud enough about just how much her asthma actually being controlled has improved her life and therefore the entire family.
Certainly keep trying docs until you find one that takes you seriously. 