I think the key issue here is if she really said on-board that she has a "life-threatening" dog allergy. Lots of people have dog allergies, less have "life-threatening" allergies and that's a serious issue that needs to be resolved before boarding.
Not saying she should be dragged off the plane but if they require medical clearance for someone saying they have a life-threatening allergy then she needs to provide it - how does she know a dog didn't sit on the floor at the seat she's in on the previous flight??
Southwest website says this:
Animal allergies
If a Customer is severely affected by allergies to an animal and notifies us of his/her allergy at the departure gate, we can ensure that the Customer with the allergy is seated as far away from the animal as possible.
Southwest is required by law to transport assistance and emotional support animals accompanying Customers with disabilities. Southwest requires that pets remain in an animal carrier throughout the duration of a flight. However, we cannot make such a requirement for assistance and emotional support animals. We also cannot require that Customers traveling with service animals provide advance notice of their intent to transport the animal. As such, we’re unable to provide advance notification if any animals will be traveling on a particular flight.