My young nephews (3 and 4) have handled and petted my reptiles many times before without issues. They wash their hands afterward. They get sick mainly from other kids or just because. Like the other posts, most likely your child caught it somewhere else. Most likely your doctor is wrong, just as they diagnose my niece that she is allergic to cats and cannot have them, but yet she is fine when she is playing with the neighbor's cats. Fear can overshadow reason.

However, if I were you, I may probably do the same. We don't have kids but my fur pets, especially my cat Happy who is my heart cat, when she was mysteriously hurt, the first thought in my mind was to rehome my other cats who may have (But most likely not) injured her. I wanted to protect her and shield her from every potential harm. That parental instinct, I think, if you need to rehome your pets, I would not judge you for.