My son's dog allergy at 7yrs was considered life threatening. It did change our lives, however despite it being impossible to avoid it was manageable. For instance highest dog and cat dander is in school from others children's clothes. Dog allergy is not the hair as such but their saliva. So a dog licking would cause contact reaction. Inside a room that a dog had been would cause breathing difficulties. But I never stopped my son from stroking a well known dog outside in fresh air. If his hand itched a wet wipe controlled that. My son loved dogs and was desperate to have one. By 14 his allergy changed and is now low. At 23yrs he lives with his girlfriend and 2 dogs. We have a dog and put measures in place to reduce dander exposure. My husband is also mildly allergic.

Second. If they know they need to keep animals out of area. You would think they would have a dedicated area for this reason.
Heck we collect stuffed animals for sick kids and they wouldn't last us hand them out saying they could carry germs(understand). I would think same for animals.

Yes I would tell the hospital staff that he is extremely allergic and can become very sick if around dogs. Tell them when you make the appointment. Also I would call day of appointment before you get there and few minutes before you arrive. Perhaps they need to have a special room and area where folks with service dogs go. Also they should accommodate you under Americans with disabilities act. Staff need to be aware and make sure appointment isn’t made when someone with service dog around. They can put reminder notes up.

Thanks ladies for all the great suggestions. So helpful in this situation.