You can’t ban a service animal because of allergies of other customers, or employees. When I worked in a library (a military library, but like your public library), we had an employee that was so allergic to dogs that she could tell several days after the MPs had their working dogs in the library for training.
We asked about service dogs, and her allergy, and were told that if she was allergic, and a service dog came in the library, she could leave until the animal left. However, the animal could not be excluded because of the woman’s allergy. She would get breathing issues, I guess from the dander. This woman had also been a teacher for a while, and didn’t go back to it because of the service animal issues, and her allergies. She taught in high school, and the local high school had several service dogs about the time she thought about going back, but it was too much risk for her to be in the same place with them. Plus, she was worried with constant exposure her allergies might get worse.
I’m one of those people that does the right thing during an emergency, and when she started carrying an Epipen, she told me what to do if she had a bad attack. Trust me, if necessary I would have done it, but it didn’t make me want to ever need to.
Her brother’s allergies were worse, and life threatening. He went to look at an apartment, or condo, and asked about previous tenants with animals, and told them about his allergy. The landlady didn’t think allergies were real, and lied about the previous resident’s several cats. Lewis walked into the apartment, and very soon after was having a major issue, and 911 had to be called. All because someone didn’t think it could be that bad, and lied to him.
My mother had a selective allergy (I doubt she had a real one, she just didn’t like animals, and liked attention) where her childhood asthma symptoms went away after her mother got rid of her several cats. If my mother knew about a cat being in a house, she started wheezing, and teared up. If she didn’t know the animals were around, no symptoms, so that’s why I didn’t believer her.