I don't think that it is likely that mild asthma is causing shortness of breath and pulmonary hypertension. I don't know what age demographic or gender you belong to but you may be suffering from primary pulmonary hypertension, or a secondary cause of pulmonary hypertension, such as blood clots to the lungs, severe obstructive lung disease, restrictive lung disease, connective tissue disease such as lupus, scleroderma or dermatomyositis. These are in addition to any cardiac causes of pulmonary hypertension (called shunts such as patent foramen ovale, ventricular septal defect), which were presumably not seen on your echocardiogram. These comprise a short list of causes of pulmonary hypertension and shortness of breath. Cardiac causes of pulmonary hypertension are far less common than pulmonary causes, but once again I would have to take your demographic into account to estimate what are the likelihoods. You probably should see a pulmonary hypertension specialist primarily because of your symptoms and persistent tachycardia.
How high were the pressures? Remember, an echo is just an estimation, and a right heart cath is the only way to tell for certain that someone has PH. An echo can be way off. I would seek out a pulmonary guy who specializes in PH, and sometimes they can be hard to find. Not all cardiologists know how to deal with it, or even know that they don't know! It can be misdiagnosed as asthma. Hope this helps.
I don't know how high they were--he seemed incredulous and made the tech do it again and again...and then just shrugged, told me they "were high", and suggested it was all due to asthma (which I don't have any symptoms of, and peak flows are fine)
It certainly explains a lot of my symptoms of tachycardia, dsypnea, and chest pain on exertion.