Look up the IEP timeline. I think it's the same across the US. They have 30 days to reply to your request and deny or say they're moving forward with evals, 60 days to do the evals, and another 30 to write the IEP if warranted/required in your state. If your state does not require them to write an IEP (which many don't, they only spend the hours when offering services), then, you'd be done theoretically 90 days from when you make your request. I think if you move before that 90 days they'd just send your paperwork to the new district and finish it out.
Personally, I think having baseline scores is incredibly helpful. That paperwork and the diagnosis would give you access to the NLS (National Library Service), cover your butt if your state has regulations about homeschooling, etc. It would make you confident you're not missing something, like a vision problem or ADHD or... And also they would update your speech evals. A 7 yo should be 100% intelligible, even if some of the sounds are still iffy. So if in your state they give homeschoolers services, he would qualify for speech therapy. Even just having the updated evals could help you decide what to target.
My ds has an IEP through the ps. It's a huge pain in the butt and the only reason I do it is because we have a state scholarship that requires it. We had private evals, but I can tell you that the ps here can run a lot of the same tests our $$$ private psych ran. If you were to make the written request NOW, before school even starts (which you can, yes!!!), you would get the timeline running. It's their least busy time of year for running evals, because they probably just have a slight backlog from the previous school year. It's a really good time to be doing this, and they might literally get you done right away or much more quickly than you anticipate. Like maybe 60 days from the letter you'd have a report, kwim? It really could be that fast by doing it in the fall.
Our ps wanted to see evidence of RTI (Response to Intervention). What products have you used with him? The ps does not *have* to eval as a first step. They could tell you they want to do RTI first and see how he responds to that. So you want to be able to demonstrate, going in, that you've already done increasing levels of intervention. They're going to want evidence that there has been adequate instruction, and they can deny you for evals if they suspect RTI should occur first.
As far as what to do? I would give him the Barton pre-test, get a fresh speech eval to figure out what is going on with the speech, and see if you need to do LIPS or FIS. My ds has verbal apraxia (motor planning of speech problem), and we ended up needing to blend LIPS and Barton for the first two levels. We also blended into that the speech therapy methodology. It was really powerful for him. You'd like to know if his continued speech problems are from the dyslexia, motor planning, some auditory processing issues, or a combination of things. When you contact the school, you want to say EVERYTHING you're seeing to get them to check AS MANY BOXES on the eval planning form as you can.