As @kanaazpereira has said, a second opinion is a very good idea right now.
High blood pressure as you have noted, is something you need to follow up on. If you do have a rare disease, as your cardiologist has suggested, even more of a reason to seek a second opinion at a multi-disciplined medical system like university medical school or a facility like Mayo Clinic.
Have you ever had a pulmonary stress test? This is where you walk on a treadmill (or pedal on an exercise bike) while wearing a mask over your face that registers your breathing as you walk on the treadmill. Your blood pressure and heart rate are also monitored. Here is a link to an article from the NIH regarding this procedure, https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2734442/. Down on this page, there is also a picture of how the test is done.