Just returned from 2 weeks in Kauai (Poipu) yesterday. Our 7th visit to Kauai. Concur with all the Sept/early Oct recommendations. Oct is fine for Poipu but iffy for Princeville. Be aware that the landslide and road blockage between Princeville and Hanalei create considerable traffic congestion around Princeville. Last Thursday afternoon we could not get to Princeville due to the long backup on Highway 560. Only one lane is open past Princeville, access hours are limited so they can work on it and weekday backups are huge. Work is expected to continue for several months. Google it for more info.
Given that access to Hanalei and Kee beach will be very restricted (parking reservations now required for Kee) I would suggest you reconsider Princeville and spend more or all of your time in the south. Poipu beaches are much better/safer for kids and access is easier.
Also be aware that restaurants on Kauai are still struggling to get back to full service. Covid restrictions should be lifted by the time you go, hopefully. Currently masks are requited indoors and restaurants are limited to 75%. But due to staffing problems hours are often limited and menus are reduced. Reservations are pretty much mandatory at the nice places - often booked out 2 months (Tidepools and Beach House, for example). Bubba's hamburger place is closed two days a week due to "staffing shortages". Quite a few of our past favorites are just gone.
Not to say we did not have a wonderful time. Just be aware of the current limitations. we are not back to "normal".