If people say stupid things on phone calls, put it in an email to 'confirm what they said'. Either they'll have to retract it in writing and say something more sensible, or you've got a written record of them being reckless which you can forward to someone who can provide better training.
Dear PE teacher,
Re Child X asthma medicines
Thank you for your call yesterday. I am emailing to follow up on a couple of the points raised.
You mentioned on the call that my daughter should have taken her inhaler with her to the sports field. She was not aware that she could do this, I have now informed her that she should bring it and [write in here where exactly it should be stored - put it in the medications tray, or whatever else the arrangements are - if you don't know what the arrangements are you need to go back to either the PE teacher or whoever formulated the asthma plan and find out. This is the most important thing.].
You said that it was not the case that you did not allow DD to fetch her inhaler. She must have misunderstood the situation as she is convinced she did ask and was not permitted. I have let her know that while she should remember to bring her inhaler, if it inadvertently gets left behind again her safety is more important and she should ask a teacher and expect to be able to fetch it. I would of course support any negative consequence you felt appropriate for forgetting.
You also said on the call that you felt DD could not be experiencing shortness of breath because she was not engaged in strenuous exercise. I would like to highlight that exercise is not the only thing that can trigger an attack, and degree of exercise is not relevant to whether an inhaler is needed. This is explained in DD's Asthma Plan which you should have on record, I have attached a copy for quick reference.
You may not be aware that DD is new to managing asthma as she has only been diagnosed following her recent pneumonia. I would ask you to please be supportive as she gets used to managing her condition, she is less experienced with this than others who may have been dealing with the condition throughout primary school.
Thank you for your help in this matter.
cc to form tutor and whoever set up the asthma plan/a pastoral person.