Hmm, while I'm asthmatic and I avoid antibiotics as much as possible as I'm allergic to one (penicillin) and highly sensitive to another (erythromyacin) which doesn't leave a lot left
I have to say that I've had a very phlegmy cough thing for the last two weeks and it's gradually clearing and I haven't bothered going to the GP (although I have been on full dosage of preventers and relievers for the duration).
I think if there was any sign that at max previously agreed doses for preventers and relievers I was still struggling to breathe then I'd be down at the GPs like a shot, ditto if the stuff I was coughing up was in any way fluorescent (you know the type I mean!) or if I had pain in my chest.
I think if you've been asthmatic for a long while then you get to know when you're struggling and need extra help - the trick is to trust those instincts and act on them (which I have to confess, I'm bad at doing).
All that said, whenever I've been to the GP and I've asked if they think antibiotics are strictly necessary, they've been 100% honest. They know I don't like taking them lightly. And it might be that you don't need them but would really benefit from having a short course of steroid treatment or an increase of asthma meds just to get you through to recovery.
On balance, I have found that if I've needed to ask the question whether I should go to the GPs or not, I really did need to go.
Hope that helps, and I hope you feel better soon.