Why would you see a pediatrician for a phobia?
They don't treat mental health issues.
If my doc told me to do anything except make an appointment with a psychologist for the kiddo I'd change docs. This is not their field.
That said.
He truly sounds like he's dealing with some OCD. Touching an object and it making him safe or okay is very much a compulsion that he might need meds for but therapy is a much better choice at his age.
They can use many techniques to get him over this if there was a traumatic event and this isn't a biological issue, chemically caused.
Such as getting embarrassed because he threw up and everyone in class laughed verses he is just terrified of something like this happening and by walking along and touching all the handles on the lockers in a certain hall will keep him from vomiting.
This is an unusual phobia since it has nearly a zero chance of happening. Even if it did it's not the worst that could happen, I mean there are worse things a body can do in public. Throwing up is something that most would feel sympathy for and be right there to help that person. It gets empathy. Whereas something like flooding their space with diarrhea would be really humiliating and laughed at.
I think he needs to see a therapist that can work with OCD/OCPD and get to the root of this. Event or chemical issue. Then they can suggest a PSYCHIATRIST to write your child's prescriptions.
Again, a Pediatrician should not be treating your son for this in any way. Only mental health professionals should be working with your son.