Excuse the copy & paste, but the following is from the first two paragraphs of the section on clinical features of coeliac disease in a standard pathology textbook. I've added emphasis where appropriate. (It's an American book, so please excuse spellings.)
In adults, celiac disease presents most commonly between the ages of 30 and 60. Many cases of celiac disease escape clinical attention for extended periods because of atypical presentations. Other patients may have silent celiac disease, defined as positive serology and villous atrophy without symptoms, or latent celiac disease, in which positive serology is not accompanied by villous atrophy. Celiac disease may be associated with chronic diarrhea, bloating, or chronic fatigue, but is often asymptomatic. These cases may present with anemia due to chronic iron and vitamin malabsorption. In adults, celiac disease is detected twice as frequently in women, perhaps because monthly menstrual bleeding accentuates the effects of impaired absorption.
Pediatric celiac disease, which affects males and females equally, may present with malabsorption or atypical symptoms affecting almost any organ. In those with classic symptoms, disease typically begins after introduction of gluten to the diet, between ages of 6 and 24 months, and manifests as irritability, abdominal distention, anorexia, chronic diarrhea, failure to thrive, weight loss, or muscle wasting. Children with nonclassic symptoms tend to present at older ages with complaints of abdominal pain, nausea, vomiting, bloating, or constipation. Common extraintestinal complaints include arthritis or joint pain, aphthous stomatitis, iron deficiency anemia, delayed puberty, and short stature.
("Positive serology" means the patient tests positive for tTG AgA or DGP IgG [see below]. "Villous atrophy" means the lining of the small intestine, which usually has lots of projections [villi] and indentations [crypts], is smooth."Aphthous stomatitis" refers to repeated formation of mouth ulcers.)
So although she's a little old for onset of paediatric coeliac disease, and quite young for adult onset, the presentation isn't unusual and the positive deamidated gliadin peptide (DGP) IgG test would ring alarm bells. Do you know whether she was negative for tissue transglutaminase (tTG) IgA? Does she have results for human leukocyte antigen (HLA) types?
The contribution of genes to development of coeliac disease isn't straightforward, so the lack of a known family history doesn't tell you everything as previously affected relatives might have been a few generations back.
The definitive diagnosis would possibly come from endoscopy and intestinal biopsy, although this has become less common as more reliable blood tests have become available. The NICE guidelines (www.nice.org.uk/guidance/ng20/chapter/Recommendations#advice-on-dietary-management) say that the diagnosis should be confirmed by a gastrointestinal specialist before implementation of a gluten-free diet, and this is also what the NHS web site says (www.nhs.uk/conditions/coeliac-disease/diagnosis/). Ideally you & your daughter should talk to the paediatrician a bit more about this: he/she may have a special interest in GI conditions, or might already have discussed her case with a specialist, but it's difficult to find all this out in a short out-patient clinic appointment.
Ignoring a doctor's advice isn't generally a good idea. In your position I might want to know a bit more about what the advice is based on, but that would depend on whether the doctor is routinely treating coeliac patients. I'm not a doctor, by the way. I teach medical students and come across this kind of stuff while preparing medical science teaching materials.