Several questions and points of info... These are based partly on my knowledge as a long time asthma patient, and partly on my knowledge as a registered nurse (and none of what I'm about to say represents my employer or educational institution in any way, shape or form.)
A: Do you just have a rescue inhaler, or are you on a controller of some sort? If you don't like inhaled corticosteroids or combination drugs, perhaps something like Singulair (leukotriene inhibitor, pill form) would work for you. If your baseline is that low level inflammation with tendency to bronchospasm, you need to address the inflammation somehow or you will continue to experience symptoms.
B: When's the last time you had a full checkup? Have you ever been screened for cardiac involvement? Folks with respiratory issues can be more prone to cardiac problems, or you can incorrectly attribute a symptom that's cardiac in nature to the asthma.
I'm a hellacious case in point here... I have vasospastic angina (fatigue, palpitations, chest pressure, syncopal episodes, comes and goes intermittently.) I always figured the fatigue and chest pressure were from bronchospasm, and the syncopal episodes were stress induced or related to low potassium. WRONG! I went way too many years without figuring this out. Please, don't let that be you.
C: Do you have seasonal allergies? Does that affect your asthma control? If so, you might need to be addressing that as well.
D: How often do you have a flare that interferes with your quality of life or sends you to the ER? Don't downplay it... take care of it. It's great that you're trying to exercise, but you can really hurt yourself if you go into this without approaching it correctly.
Food for thought... and things to take with you to the MD office... there's no way that you should be so exhausted at the end of normal exercise that you have to lie down. That does not indicate good asthma control. If your MD doesn't listen well or you feel like you can't communicate with them, find one you can communicate with. That can make *all* the difference.