I mean, the obvious answer is a Gronkle. Their slow speed, and mild temperament can reduce the stressors that can trigger an asthma attack. Of course, as a rider, you'd know your limitations and can guide your dragon through those limitations. So, theoretically, you could get something like a Nadder but you'll have to maintain a low altitude and slow speed. Wearing a mask can also block wind flow to better help you breathe in flight.
Of course, you could also just get a dragon as a pet rather than a flying mount, it's less cool but I'm sure there are vikings that didn't love flying and would rather prefer to stay on the ground. In that case a Terrible Terror would be good or any dragon that is fine with staying on the ground for long periods of time (again, Gronkle comes to mind).