My son was negative on the most common Celiac blood tests but was positive on the two that dealt with IgG (rather than IgA). If he had not been given the complete panel of tests, we would have missed it. He also had no classic symptoms. The doctors were puzzled, but we got our answer when he had a "highly positive" biopsy after an endoscopy. His symptom (erratic behavior changes) went away within days of being gluten free - it was amazing.
Your child has a lot of the classic symptoms and it looks like you only got a few of the possible blood tests.
If your doctor does not want to test further, you can always just try the diet and see if it works. Our doctor considers that the "final test" for kids anyway, since the other tests are not always accurate with children. If you see improvement, stick with it.