She sounds very much like my sons. My older son also started falling off the charts (despite starting out above the midrange) at about that age. By age 2, the pediatrician said that he was just going to be petite. Well, the pedi was totally wrong and there was something wrong with my sons. The fact that your daughter is consuming lots of food but not gaining and has a belly are all signs of celiac/gluten intolerance. My sons weren't diagnosed until they were 7 and 10 when they stopped growing altogether. After we started the GF diet, my oldest son grew 4 inches a year and gained 20 lbs a year for over 4 years. At age 14, he was 6' tall and now the child the pedi said would be petite is 6'4" tall and weighs almost 200 lbs at age 17. My younger son lost inches in his waist after starting the GF diet - that was because his gut wasn't fat - it was the fact that his intestines were inflamed from the gluten.
The fact that the doctor is encouraging a high fat diet is yet another sign as kids that have undiagnosed celiac have fat malabsorption due to the damage to the intestines from the gluten. One easy way to see if this is a problem for your child (and for all the other mamas with small kids) is to see if your child's stools float in the toilet. I didn't know that wasn't normal until I was 42 and started a GF diet.
The meat is probably very hard for your daughter to digest which is why she has an aversion. Folks with celiac often have lower than normal levels of the enzymes and bile necessary to properly digest proteins. I too had a strong aversion to meat and ate almost vegetarian until after I went on the GF diet.
This is a genetic disease and is very common in folks of Northern European descent, although anyone can develop this. This is partly due to the fact that the wheat has been so hybridized that it's not the same wheat our ancestors ate. The amount of gluten is significantly higher and our bodies cannot properly digest it, thus the amount of proteins that can end up in the blood stream often lead to this autoimmune reaction. The good news is that it is very treatable by dietary changes alone. This is really why most doctors are clueless about it - there is no Rx they can give, so based on their business model, there's no motivation to test for it. You can do the testing on your own through enterolab: www.enterolab.com
If your doctor is willing to test, then that's great. You might also ask that they measure vitamin D, vitamin B12, ferritin (iron storage) and folate. Low levels of any of those are also signs of undiagnosed celiac. If your kids (under age 6) have low ferritin, then that means mama likely does as well since kids get an iron store at birth that should last until they are around 6 or 7 (another symptom I had was that I was anemic at age 6).
I learned all of this the hard way and I would hate to see others children suffer like mine did. It's hard for kids this age to tell you they don't feel well or that BMs hurt. But, my sons were amazed at what it felt like to feel well after they went on the diet. They were both used to severe intestinal cramps (as was I) along with every BM. I never would have guessed that I could feel so well. I felt better at 47 than I had ever felt in my entire life.