Her weight is just below the 25th percentile and her height is just below the 5oth percentile for her age. It can be very frustrating when your child has no interest in food. Suggest replacing her milk at meals/snacks with Pediasure, or milkshakes (milk and ice cream blended), or yogurt smoothies (yogurt, ice cream, or frozen fruit-if he accepts it- blended) for additional calories and protein for weight gain. Recommend giving her a chewable children daily multi-vitamin due to her eating habits. Try engaging her in the kitchen more. Have her help you with planning and preparation of food, and setting the table. Hopefully this will increase her interest in food. The most important is to relax at meals. If she will not eat a certain food not eat too much, do not make a fuss over it. Let it go and enjoy the meal. Suggest asking the Expert Child Behavior forum for additional tips. Thank you for your question.
"She avoids foods as if she is trying to save her life."
Maybe she is trying to do just that. Children do not starve themselves out of orneriness. She could have a severe food intolerance (as well as allergies) which the body sees as a poison. Some people are intolerant to the nightshades (potatoes, tomatoes, peppers, eggplant), or gluten, or lactose, among others. I would investigate this area.
If she is suffering sleep apnea then she is not getting the rest she needs, further adding to her problems. My second daughter suffered sleep apnea as a very young child. Unfortunately we did not discover her problem until many years later and just thought she was like her father and didn't need much sleep.
You only see this as a psychological or behavioral problem. You should consider other possibilities. If it is physical, all the psychology in the world won't help this little girl, and ignoring the problem will harm her.