This is not to dissuade you from mediation and trying to find the best placement for your child. Each child is different and what is the best available path for one is not for another. It is to give you a different perspective.
We did not go through mediation and chose a different route. One of the reasons was that we determined that the private school options were not really better overall than our public school. There are very few private schools that will take on a student with severe LDs- there are many who take mild to moderate cases. At the time we looked (DC was also 7 and we were frustrated with his progress.), only the Lab School took the more severe cases. This was ~7 years ago and a few more schools have started taking more severe cases and a few more schools have been started. The Lab School is very small. At the time, there were about 12 students per grade and the only openings would happened were when someone left. Plus, they wanted to maintain as much as diversity as possible- gender, racial/ethnic, severity of LD and type of LD. 2nd and 3rd grade is a big time when people apply for their child. The chances of acceptance can be slim (he was the most likely applicant- white male in 3rd grade). At the same time, for us, it wasn't a good fit because they do an arts based curriculum and DC is a science and math kid and loathes almost every art activity. FCPS is the place to be for a science and math kid. We decided, for DC, to have him stay in the school where his assets would be addressed and some of his deficits and we would supplement at home with private professional intervention and parent intervention (eg. I learned the Wilson reading program to supplement what he was getting at school and through his reading teacher.) Warning: This path can be more stressful on the parents and requires more one on one parent time and schlepping time.
Can your DC access the curriculum with audio materials and by dictation?
My DC has multiple severe LDs and no intellectual disabilities nor ASD. For accessing the curriculum we used audio materials, read it to him and had him dictate answers. At the same time, we concentrated on bringing his writing and reading skills up.
Also, one of the reasons children fall back in grade level reading (or other measurement) as they progress from K to 3rd grade, is that the range of what is considered average is quite wide in K and narrows as they get older. They may be making progress on their own timeline but be not as fast as their peers are progressing (it is a time when NT children are making very rapid progress). Is your DC making progress in reading? Our DC had a similar drop and then the gap leveled off, but he was always making some progress.
He is currently in HS. He uses audio books and technology to read pdfs and word documents. He uses a word prediction software program for writing and we rarely use dictation anymore, except in special circumstances. He is still experimenting with speech to text software and hasn't found one that works well for him. He will be taking his first AP class this fall.
Good luck with your DC and let us know the outcome!