JesusS is wrong. "El señor pasó a ser veterinario." we don't know what he was before.
I have no issue with using "ser" twice. I like it. Oor don't use it at all: Chuky pasa de muñeco normal a muñeco asesino.)
There is a degree in changes.
For instance, Transformó implies a deeper change, in something so strong that you can feel you can grasp it. So it is normally a change in shape. If you say Chuky se transformó , I picture his face changing into a murderous one.
Convirtió is like transformó, but less strong. So Chukey acquire the attitudes of a killer. He was mean, and perhaps his face changed.
Pasó a ser just implies that he played the role. For instance, Chuckey killed by accident, and know he is called a killer. It is true that this meaning can be used for "convertir". But still convertir can go beyond in most cases; some sort inner change. Once more, this is no the case of "pasar a ser". Actually, in example above, paso a ser veterinario, implies he has no studies on medicine or animals, he just somehow manage to get enough knowledge. However, convirtió implies a process, perhaps no schooling, but he went trough something.
Finnaly, pasar a ser is also very unexpected, almost magical. You can say: La oruga se convirtió en mariposa, but you cannot say pasó a ser mariposa.
"Chuky pasa a ser" describes better the change, since it was magical, and it was not a process, but a change from one scene to the other.
"Chuky de convierte" means that there are real, complex and interesting thing going in this change. You can even get to think that it took half movie for this change to take place.
There is no negative connotation in convertirse. Go ask the butterflies.