I think there's a slight difference, but I don't think I could explain it. First of all, it seems to me that the sentence should read, "... befitted someone who ......" or maybe "...befits ......"
Saying "..... as befitted someone who ........" seems to be explaining why he might be working well with children, and assumes he's already working with children. Whereas "....... which befitted ......" could mean that he isn't working with children yet, but that the narrator of the story believes he might work well with children.