I don't think the issue is irrelevant, because there is a history of something. I cannot give you an answer, but I can just ask the same questions you already know you are going to have to answer.
How are you going to answer these questions on the health history form, Have you ever:
Consulted, or been treated by clinics, hospitals, physicians, healers, or other practitioners for other than minor illnesses?
Had any injury or illness other than those already noted?
How are you going to answer, Have you ever had or do you have: Asthma?
Answering yes to any of these questions is going to require an explanation of the incident, illness and treatment.
Are you going to answer no, and say the difficulty breathing was a minor illness or no illness at all. I don't think it was a minor illness and something that is irrelevant.
Do you know what the specialist diagnosis was? I would think that they picked something to put in the records to justify the inhaler prescription. And my guess is that it was Asthma. But, maybe you don't want to know the diagnosis, because it may have been exercised induced asthma, along with the use of an inhaler (even though it had no effect). If you were diagnosed with asthma, and you check yes to asthma, you will be going down the asthma rabbit hole. I think you need to know what the doctors diagnosis was.
In the DoD medical standard http://www.dtic.mil/whs/directives/corres/pdf/613003p.pdf
they talk about "reliabily" diagnosed Asthma. They define "reliably diagnosed" as: "Reliable diagnostic criteria may include any of the following elements: substantiated history of cough, wheeze, chest tightness, and/or dyspnea which persists or recurs over a prolonged period of time, generally more than 12 months." Based on this definition your DD asthma (if that is what it was) was not reliably diagnosed, but that determination is not for you to make, it is for DoDMERB. DoDMERB may be inclined to DQ for asthma and then send this information onto NROTC for them to make a waiver determination.
Make sure you look at the medical standard linked above. Either search for asthma, or look on page 17. However you answer, make sure you put in the explanation what it was not, or give DoDMERB an easy road map to make their determination. Maybe it is something like this: On xx/xx/2016 I had difficulty breathing during a soccer tournament. No immediate medical attention was required and I continued to play in the tournament. On xx/xx/2016 I went to Dr. XXX, but there was no reliable diagnosis. The condition resolved with no treatment and I have not had a recurrence. After I wrote this hypothetical response, I don't know if it triggers more questions, or a remedial to get the medical records to see what it was. I just keep rambling, and struggling with a "proper" response.
Just remember, my response is from someone on an anonymous forum with no formal medical training. Take it for what it is worth. And most importantly answer truthfully.