Hi Blu3,
I'm not sure if you've spoken to a doctor from AVMED directly or not, but if you have you can be fairly sure that their advice on medical rulings will be sound. Don't listen to DFR - they don't know much on the matter. Only listen to the doctor.
Essentially, for every cross on your class 4 medical, you'll have to see a specialist to confirm that you're a low medical risk for flight training. I had a suspected migraine when I was 10 and had to see a neurologist to get the thumbs up. In relation to asthma, you'll be hard pressed to get anywhere with an appeal without having done a challenge test administered by a specialist. Your GPs say so won't be enough. The challenge test is similar to a spirometry, and as mentioned before the exception is that you inhale a powder designed to induce an attack. Everybody will have some reaction to this powder, however if it falls outside a certain range you're deemed to have active asthma.
This was the exact process I went through and they seemed to take a fairly hard line on it, but once it was done there were no questions asked and I was given the all clear.
What confuses me about this though, is why have you been given a class 4? I was under the impression that should the medical component of assessment day not go to plan a candidate would be given a class 3R until specialist testing was undertaken to determine the medical risks associated with the relevant condition. You obviously haven't undertaken any specialist testing in regards to your asthma, so why the class 4?
I'm just a layman here, so if I've got out of date info someone please feel free to correct me.