1- Is the motion of her esophagus normal? A known complication of esophageal atresia surgery is trouble in the motion of the esophagus which makes the swallowing painful and difficult. Xray of the esophagus during swallowing will show if this problem exists.
2- any residual reflux? If your daughter spits a lot then she's not keeping down enough food. Also if she chokes frequently, she may have small food particles going into her trachea (micro aspirations) causing inflammation +/- infection.
3- any strictures (tissue that grows during healing after surgery) that caused narrowing in the esophagus?
4- any associated heart or kidney disease (this is common in esophageal atresia, she needs heart and kidney echo to be sure)
5- was she investigated for thyroid problems or gluten sensitivity (celiac disease) which very often cause failure of growth?
I urge you to consult a pediatric gastroenterologist and a pediatric nutritionist to do the necessary investigation and give her the right diets fatty food is not only not enough, it may cause diarrhea (poor absorption of fat) and later on heart and blood vessel disease.
so if your dr advice you to give her oily food then this must a temporary solution to let her feed but not the appropriate long term treatment ,your girl must be followed up by a specialist to assess her condition clearly and make the necessary ,if the cause of the problem is strictures this can be solved by another operation ,and be sure she is still not having acid reflux that may worsen the condition through taking the necessary medicine that is suitable for her age
so , look for the long term solution for your kid 's problem not for the instant relief of symptoms