I've had multiple panic attacks like these in the past and I've still not found a way to prevent them. The worst time it happened I remember begging my parents to take me to the hospital because my throat choked up and I could barely breathe, just like with your case. The only thing that got me through was my Ativan, which is a short term anxiety reliever. It lasts a few hours and the only side effect it gives me is drowsiness. I know you mentioned not wanting a medication, but you only need to use it when it's needed. Also, my pills are so small they're practically chewable, which also helps when you can't breathe. Sorry if this isn't the answer you were looking for, but it's the only solution I've found.
Short term medications will not mess you up but they can be addictive if you are careless and take them too often. You need to work on what causes the panicky feeling that you can't breathe, but realizing it is all in your head is the biggest part of the battle which some people just can't get over.
You can try therapy or maybe clinical hypnosis would work since you already know what you want and are open to a solution.
ST meds might be necessary to give you a break once in a while if the therapy solution takes a while, but see what doc thinks.
With panic attacks - you need to learn about how they occur. I worked in a anxiety clinic and that's what we did with clients; This is quite blunt, but the idea is to see how panic attacks will not cause you to die and that the symptoms - so not being able to breath, pressure, faintness, etc. can be triggered easily by yourself and can also eventually be stopped. Have a look at Clark model panic attack. CBT for panic attacks work really well and it is quite fast - if you can't afford a therapist, self-help CBT for panic attacks work well too. You can also try relaxation techniques, such as muscle relaxation, mindfulness, breathing techniques (those are quite useful). I hope that these aid you as a starting point! Also, don't worry to much regarding the medication, I haven't heard of them making people worse. That said, if you are worried about that, talk to your doctor :)
You say no breathing tips help, but I doubt you've exhausted the universe of breathing exercises -- they come from Eastern meditative practices and there are hundreds of them. A psychologist who specializes in anxiety treatment should know several, but if you go to, say, a Buddhist teacher to learn Mindfulness meditation, one that has very good clinical results with many people with anxiety problems, that teacher can also teach you a bushel of breathing exercises. But a good CBT practitioner will usually know several, as will a therapist who has studied hypnosis. If you see a therapist who doesn't know any of these, that person doesn't specialize in anxiety treatment. As for medication, try therapy first if you don't want meds, and see if it helps, though it will take time. Your general doc probably won't be of much help other than to give you tests to see if there's some underlying physiological problem going on, such as thyroid or blood sugar problems. But assuming it is an anxiety problem alone, regular docs will only be able to give you meds and they're not that well trained in them, so if you do decide to go on meds you should see a psychiatrist. Good luck.
Things that have helped ease my panic attacks:
Deep focused breathing
Being mindful of surroundings to take focus of myself
Also raising blood levels in iron, vitamin D, magnesium and Vitamin B12