My child was diagnosed with cough variant asthma but is it more driven by inflammation vs a regular?

No... Cough variant asthma means that the inflammation and bronchospasm results in coughing as the predominant symptom instead of wheezing. Other than that, cough variant asthma is still asthma.
Inhaled steroids. Cough variant asthma caused by inflammation is best treated by inhaled steroids. Check with your medical provider to determine the best one for your child.
Emf. I found kids who play video games developing cough variant asthma - electro magnetic fields by tv, video console controllers - we used emf guards like qlink pendants, we q taped the fuse box at home, rectified biomagnetic fields around house with magnetic hematite, for kids orally did limonenine 1drop with vitamin d3 the coughing that was present for 9 months stopped - think about emf fields!