After talking with some other service dog handlers, I decided the best way to deal with the rheumatologist who is afraid of dogs was to leave Isaac at home for appointments when I can and take a friend with me instead. However, that probably will not always be possible.
So, also on the advice of other service dog handlers, I decided to call her and ask what I could do to help her feel more comfortable with him present. For instance, I could sit on the far side of the room from her, as long as she was not physically examining me. At my first appointment with her, she did examine me, but last time, she just sat at a desk and talked to me. The medical assistant took my vital signs before the doctor came into the room. I could also bring a portable crate with me and crate Isaac while the doctor was in the room. They make soft-sided crates that are lightweight and fold up to about the size of a diaper bag. Or, if the doctor had any other ideas, I was open to hearing those.
Some people suggested I just fine a new rheumatologist who is comfortable with my service dog, but I live in a very rural area and this happens to be the only rheumatologist within a 45 minute drive. There is another about an hour away but he does not take my insurance. So I'd end up having to drive more than an hour to see someone else. Which I could do, it's not like I see the rheumatologist very often. But I'd prefer to keep seeing her if possible.
So I called and spoke to the medical assistant, who then talked to the doctor and called me back. The doctor said she would "try to work around my service dog" and let me know if she decided she was just too uncomfortable. The medical assistant said she thought crated Isaac during the appointment would help, so I will do that.
I'm not quite sure how I feel about all this. I understand some people are just scared of dogs. I don't know why she is scared of dogs, but I don't really need to know. Maybe she had a bad experience in the past. I want to respect her feelings. I'm glad she is being honest with me about it. And she didn't say I couldn't bring him, she's not violating the law. It troubles me, though. Hopefully I can get a friend to go with me to my next appointment.