No BTDT advice...but from my reading of the admissions criteria, it appears that you would not need any new testing beyond what you already have up to and including the 2022-2023 admissions cycle.
You have several options for exploring admissions, I think, including 1) just starting an application for this cycle and letting it play out (remember, you are not required to disclose a disability); 2) inquiring about disability services, with or without starting an application; 3) unschooling through the rest of this year and next, and then applying for the 2022-2023 cycle--but that just puts off the decisions about 1) and 2).
But there are a few other points of context. Firstly, no public school (and it is a public school) is allowed to discriminate against students on the basis of their disability, even if they are a selective school, if they otherwise meet the admissions criteria. You have qualifying test data in hand at this moment, and once he is in-state, he will meet the residency requirements for the day school. If you are concerned about the "middle school level" academics statement, remember that he would have completed a standard fifth-grade curriculum (at least) by the end of this school year, which makes him minimally middle school level by definition. I think you're in the clear on this one.
I would be less concerned about not being given a chance to be admitted, and more concerned about whether they successfully serve 2e students (I know nothing about them in this area directly, so I'm certainly not suggesting that they do a poor job at this; it may be excellent). It may be helpful for conversations you have with them regarding 2e-type concerns to be specific to scenarios described more in asynchronous terms, initially, if you feel uncomfortable with disclosing a disability prior to admissions--although that may be somewhat unavoidable, since I assume the diagnoses are written on the psych evals. Or you could start with open-ended inquiries into diversity in the school population, of various kinds. They may volunteer the information that you are looking for in a more spontaneous way, which may give you more confidence in how consistently the environment embraces students like your child.
And fwiw, I'm guessing some variation of your DC's particular quirks are probably not that uncommon in the population the school was designed to serve. There haven't been behavioral or emotional concerns significantly out of the range in a standard school setting, so there's no particular reason that they would suddenly arise in this new setting, or to anticipate that recommendations or school records from previous schools would cite them as a caution.
IOW, I wouldn't weight admissions chances too heavily into your schooling decisions for the next few months or a year. Do what makes the most sense for DC's needs now, and apply (or don't apply) on the schedule that fits his needs and your family needs for the coming school year.
