The respiratory and cardiovascular systems are closely linked and changes in one of them provoke changes in the other. Why do you need to improve your breathing apparatus?
Is sport obligatory? Try playing a wind instrument (flute, clarinet). I know from myself that it develops the respiratory system very, very well. Better sports. I have never had shortness of breath while running and other serious exertion (and I did have problems with the respiratory system). Yes, and there seems to be no load on the heart, besides, music, on the contrary, soothes. This is the best option.
And, alas, I cannot give specific advice on sports. Any sport involves stress, and any stress affects the heart. It is better to contact a cardiologist who will recommend something suitable for your situation (after all, heart problems can be different and for each you need something different, maybe you need light cardio training, or maybe it is contraindicated).
The problem is that breathing quickens due to lack of oxygen. And there is not enough oxygen due to the rapid heartbeat (a rough example without details). That is, one cannot exist without the other. Consider swimming, running, crossfit with minimal loads, but only after confirmation from a cardiologist exactly for your situation, but better learn solfeggio :)