Sept will likely be hotter, more humid, and more rainy - and chance of hurricanes. But, I would wager a guess that the crowds will be lower. There are schools that get out at the start of May around here - though far more don't until mid-May. I would think that Sept will be low crowds and the week in May you're considering would be low-to-moderate crowds as long as you're staying away from Star Wars weekend and not getting into the third week of May.
MLK weekend has much higher crowds. And though I think Sept is your best bet for lower crowds, I wouldn't do it if meant pulling both out of school. I'd go with May and enjoy the lower crowds at the start of your week and have a plan to deal with any higher crowds at the end of your week.
Make sure the child who will miss school checked with the school/district. There are some very strict ones - like my state will not graduate a child who misses 5 days(block scheduling) of school in one semester in high school unexcused. (Vacations are unexcused. Even illness is unless there's a drs note. This is a state law, so our public schools don't have wiggle room). Even if your state isn't strict, pulling for high school is a major thing, even when finals are not involved. I would have not wanted to miss a week of school at that age. So discuss it with the child too, not just the school. Otherwise, go as soon as the high school year ends. It will be moderate crowds, not high then, as many kids are still in school in June.