Yes, I also was diagnosed with exercise-induced Asthma, and like you, have a lot of coughing at night. Haven't found out for sure if it is cough-variant Asthma, but it sure feels like Asthma (chest tightness, shortness of breath, feels like something is constricting the ability to expand my lungs, hawking up lots of flegm) and I don't have any audible wheezing when I feel this way.
I also get the sore chest & fatigue, so yes I think that is part of the Asthma. Might be good to ask your Doctor about the Cough Variant form of Asthma, and also to recommend something to keep your Asthma from getting out of hand at night. I'm getting a nebulizer soon to use daily as preventative against the nightly "attacks."
Good luck & take care,
Gluten & Sulfite Sensitivity, Many Food/Inhalant/Medication Allergies, Asthma, Gut issues (dysmotility, non-specific inflammation), UCTD (Lupus?), Osteoporosis, Anemia, Very Low Lymphocyte Counts (T-Cells & B-Cells), malabsorption/malnutrition, possible Lyme Disease (Igenex Lab IgM WesternBlot positive), etc.
Meds: Pulmicort, MaxAir, Injectable Vitamin B12, Herbs, Nutritional Supplements, Essential Oils, Homeopathy.