The result of all this was that he is most likely fine! Apparently breastfed babies often grow slower at this point in their development and he may just be a small baby. She advised starting him on solids right away, which we were planning anyway, so last night he had his first try of proper food. (Well, I wouldn't count baby rice as proper food, but then I'm not a baby!)
Tomorrow we'll try him on puréed carrot and see how that goes!