The key words: "She is starved all the time."
Sounds exactly like my 9 year old daughter. She, too, is big for her age. In her defense, she is built like her Daddy (think Scottish Highlander build; she'll be almost 6 feet when she's grown, and she will never be a "skinny Minny.") At the moment she is 4'8" tall and weighs 125 lbs.
First, she has Sensory Processing Disorder, which is on the Autism Spectrum, and she also suffers from hypotonia and spatial awareness problems, so like your daughter, getting her to do exercise or learn how to ride a bike without training wheels (yep, still there) is a constant challenge.
She does like to swim, so in the summer I buy a family summer pass at the pool by her school, where I've been trying to teach her to swim, float, etc. (she was in swimming lessons but the noise and the other kids kept distracting her, plus the teachers wouldn't/couldn't take the time to coax her to do something. It had to be RIGHT NOW, whereas my daugher needs time to calm herself and work up her courage. Just since last summer she can dog paddle and pick up items off the pool floor. For her, these are big milestones.) We're doing swimming again this summer. I went as far as a certified Jr. Lifeguard, so I know all the swim strokes. I haven't necessarily seen her lose any weight from swimming, but she's getting physical exercise and activity, which is just as vital, if not more so.
I started going to an Integration Doctor for my own health reasons in Aug 2009, and I have been having my daughter tested as well. She has heavy metal toxicity; she is on her 3rd chelation. Heavy metal toxicity causes the body to not be able to absorb vitally needed nutrients, such as minerals, vitamins, water, enzymes, amino acids, etc., etc. When the body suffers from malabsorption, even though you're eating and in many cases, eating a lot of food, because the body is unable to absorb what it needs from the food eaten, the body will always register "I'm hungry" when in fact, you've eaten more than enough.
We switched over to organic/whole foods, and I've noticed that since my daughter's been chelating and has been eating organic that the "I'm hungry/I'm starved" daily, hourly, minute sayings have almost (totally) gone away within the last 6 months. I also had her nutrient panel done, and she was delinquent in quite a few key vitamins and minerals. Since she's been taking supplements, things have been better.
I had myself tested for food intolerances (I have 22 of them) but haven't found the money yet to have my daughter tested. That is another thing you might want to consider. If your daughter's eating foods that her body isn't able to tolerate, that will also make her overeat (for some reason, even though a food is "harmful" to our body, it's that very food that our body hungers for and craves the most).
After both I and my daughter tested very high for heavy metals, I had both of us take a genetics test. We both are POOR METABOLIZERS, which means our bodies are unable to completely digest and use medications and cannot metabolize any and all toxins. That means we not only have to do body cleanses and toxin cleansings, but we have to also take daily preps of special natural meds or herbs so that we can help our bodies detoxify.
I don't know if I'd totally classify your teen as stubborn. Stubborn, maybe, in that she doesn't want to exercise. But when one doesn't feel good or doesn't feel energized, it's awfully hard to get motivated to exercise. Many people are unaware just how much food and food allergies and food intolerances impact our mental health and also our waist lines. I urge you, as her caring mom, to find out if underlying food problems aren't contributing to her weight gain. My Integration Doctor had me do the ALCAT Worldwide Optimal Wellness food intolerance panel; it cost me $199, which is why my daughter hasn't been tested yet.
Another very, very missed and undiagnosed condition is Celiac disease. One of my food intolerances is Gluten Intolerance (gliadin protein). I did go on a 5-month gluten-free diet (no wheat, barley, rye, oats, or anything made out of them or any of their derivatives), but was still having troubles. I am currently on my 5th week of a GLUTEN CHALLENGE; yesterday I went in and had the Celiac Panel redone (labs), and next week Weds I go to the hospital for a colonoscopy, an endoscopy, and a biopsy to see if I suffer from Celiac Disease. I am looking forward, big time, to being able to go gluten-free again. I have been watching my body slowly deteriorate, week after week, from eating gluten. I'm glad my husband and daughter have seen how sick I've been. They didn't take my "gluten-free diet" very seriously before; this time, there will be no cheating. I'm too damn sick to ever want to eat gluten again.
So in one week I'll know whether or not I have Celiac disease--or just a very, very severe case of gluten intolerance. If it is confirmed that I have Celiac, then my daughter will get a genetics test done. If it shows positive for the Celiac genes, I will also have her undergo the scopies and biopsy. Like me, she has way too many of the symptoms of Celiac.
I hope some of what I've written here is useful to you. I've been sick since my daughter's birth in 2000, and it's only been since Spring 2008 (after the PCP I'd been seeing for 10 years kicked me out of the clinic in Spring 2007, because he couldn't find out why I was getting so sick all the time and told me to go and find someone else who could help me) and specifically, Aug 2008, when I took my health into my own hands and began seeing an Integration Doctor, that I'm starting to see a rainbow at the end of all my trials and tribulations. And I wouldn't be this far without struggling for doctors to listen to me and pushing and pushing and pushing for answers.
Good luck to you and your daughter. Please, keep us informed as to how she's doing. I, especially, would be interested to hear if she's suffering from any of my above suggestions.