It’s shocking how many people are suggesting lying in a way that’s so easy to get caught. “Weird I just took a picture with the nurses and the other doctor.” That’s going to make it even more awkward.
If I were you, just suck it up and take the picture, and then say you dont hand out your private number to patients and like to keep the relationships professional. This is presumably honest.
Not taking the picture is really spitting in the guys face. It’s so quick and it goes a long way to making them feel good, and feel good about you. It’s one of those things I would explain to my kid that you just do it and get it out of the way even if you don’t like it.
Not giving our your number is entirely reasonable, and I suspect is also honest.