I have had this for 10 years now. No doctor has been able to diagnose it or help me at all. I can only give my experience and thoughts.
I am fairly certain it is the hiatal hernia. You sound healthy otherwise, which means it is probably a mechanical issue. It is pulling your stomach into a position where it is blocking your lungs. The reason the Buteyko breathing helps is because the movement of the diaphragm is helping your stomach drop down, creating space for your lungs. I guess it would be classified as a "sliding" hiatal hernia as it moves in and out of the esophagus, which is why it comes and goes.
I got mine 10 years ago from deadlifting. Put the bar down, and suddenly I couldn't take a deep breath. Went to multiple doctors and none of them could give me an answer. 5 years later I looked into hiatal hernias and it seemed the most likely cause. Got an endoscopy and they found a small one. Doctors have been dismissive of it being the cause, but apparently the stomach being out of position by even a small amount can cause numerous issues. Also I have been experimenting with different stretches, exercises and diet changes and I am pretty sure it is the cause.
I wish I could give you some hope, but I am still trying to figure it out as well. It seems to be uncharted territory. I've heard some say it can only be fixed through surgery, and others who say they managed to fix theirs. I don't know. But definitely look into it. it could be a number of different things but a hiatal hernia seems to be the simplest explanation.
From my experience: watch out for fatty foods (burgers, oils, butter, nuts, nut butters, etc) as digestion takes longer, leaving the valve at the top of the stomach open. Experiment with different exercises and see what triggers it (for me, squats and lifting things over my head triggers it). BE SURE TO BREATH OUT WHEN LIFTING- that's how I got mine, I was holding my breath while deadlifting and the pressure formed the hernia. Grunting/forcefully exhaling air helps. Leg raises while lying on your back, as well as Windshield Wipers (watch your lower back/be careful) seem to be effective ways to release the stomach for me. Apple cider vinegar mixed with a small amount of water also helps. That's just a few of the things I have found that help me.
You can find a lot more info around the internet. There is also a book by Theodore Baroody on hiatal hernias on Amazon.
Definitely something to look into. Hiatal hernias are apparently common and manifest in people in different ways. Hope this helps.