No, dog allergies are not typically life-threatening. Giving your dog small doses of baby allergy medications (approved by a veterinarian) can help the allergy problem within 2-3 weeks.
I am not an animal doctor but my poodle also has bronchitis and coughs alot. There are drops from a vet that can help their chest. I used and it made her breath better and cough less. No do not put him down.