An IEE would only be appropriate if the school already evaluated her and found no deficits in the areas that you are concerned about. Then you would "disagree" with their evaluations and request an outside professional to do the testing.
While this is a little different than the question you asked, when it comes to reading, phonics, fluency, etc. I've found that the curriculum and the actual assessments the school uses can be the problem. Many schools use a balanced literacy or cuing approach that focuses on learning whole words, trying to "guess" words by the first sound or use pictures in the story for the context. While these strategies work for some students, the vast majority do not learn to read this way and need a structured literacy program that focuses on phonics and decoding words. I heard a great quote on this today. It was "if you teach a child 10 words, you teach a child 10 words. If you teach a child 10 sounds, you teach them how to make over 26,000 words." I've been able to successfully advocate for reading goals and interventions in students that weren't making grade level standards by using data that school already had, i.e. standardized test scores, grades in reading, etc. Your child is already eligible for an IEP so now the school has to address all areas of need, not just the areas related to the disability category. Show the need.
If you want to learn more about literacy and the different curriculums I mentioned, I would highly suggest you start by listening to the podcast "Sold a Story." It was extremely eye opening for me and I saw it first hand in my youngest son.