The school district would definitely not need to re-administer the WISC-V specifically, as it would not be a valid administration for at least 24 months after the neuropsych. They might want cognitive assessment, but there is no particular need for them to obtain another one, unless they believe this one was not accurate (which, btw, it probably is not, based on the behaviors you report--but that doesn't mean a second cognitive assessment would be any better, at the moment). If you submit this assessment, they may accept the data in place of their own. If the school requests cognitive assessment of their own, make sure they have access to his history of prior testing, so they don't waste his time and their own on invalid retesting with the WISC-V.
The additional testing the school might seek would be in academic achievement, and possibly emotional or behavioral, depending on what the behaviors of concern at school are. I would note that, if his cognitive testing is at all representative of his true abilities, I would expect educationally-relevant concerns to lie in the mathematical or written expression areas, rather than in early literacy skills, which (as has been documented) would be expected to be an area of strength. At this early stage of formal education, expectations are low for those skills, so it may be that, even if there is a genuine underlying learning challenge, it will not yet be evident in his academic performance.
If his challenges at school truly are restricted to those related to ASD and ADHD, a 504 plan may well be appropriate at this time, though one would want to monitor academics closely. If his strengths are mainly in reading, and non-desired behaviors are associated with instructional misfit, then you may be able to argue that his 504 plan should include adjusting his reading materials to higher levels as a strategy for preventing emotional/behavioral dysregulation.
