What's the solution then?
2 people with dogs, pre-advised, pre-booked, maybe even pre-paid depending on the airline policy. Boarded, seated, animals secured according to policy, customers at ease and settling for the flight.
1 person who informs the crew last minute she has an allergy and as it is not possible on an aircraft to prevent allergens from the dogs from mixing with her air, there is not much that can be done to create a "sterile" environment for her. She has no medication, the crew don't have medication for her either. She refuses to leave the aircraft and causes a scene. Even if she didn't cause a scene the airline would continue knowing that at any time she could suffer an allergic reaction and cause a costly, time consuming divert or other emergency procedure.
How many times are the staff supposed to ask nicely before action needs to be taken to remove a problem passenger from a plane?