I haven't had direct experience with this, but a close friend has. Her oldest daughter went through the very same symptoms and she had strong concerns because although her daughter ate like crazy, she wasn't gaining weight. She NEVER had a solid stool since birth it was always diahrrea, and after many adjustments to her diet, the problem remained.
Unfortunately, her pediatrician wrote off her concerns ("inexperienced new mom", etc.) until the child was about a year old and had gained very little weight over her first year. In the meantime, my friend researched everything she could find on the subject. Low and behold, she came upon information about a GLUTEN ALLERGY -- upon having her tested at Boston Children's Hospital, it was confirmed.
Gluten is present in all kinds of foods -- not just wheat products -- so it's important to have guidance in this area. Subsequently, the child's two younger brothers and eventually her own father were also diagnosed. Although their mom is not affected, she is a SAHM and does all of the grocery shopping and meal prep and is meticulous about following the diet exactly. Needless to say, this solved the problem and although it can be a lot of work to follow, there are many more options out there in terms of gluten-free breads (rice flour is OK) and even baked goods that the children can have. My understanding is that the purchase of these special foods can be used as a tax deduction as well.
In case you're wondering, she did change pediatricians and is very satisfied with the level of care and guidance she receives from her present doctor.
I hope this helps!