For the period that you have a car, you can be very flexible. If I were you I would only make hotel reservations for the following dates
1. Saga since you know that want to attend the festivals, and that weekend is a public holiday weekend so it may be crowded.
2. Fukuoka since your arrival and departure dates are fixed.
3. Amami Oshima since that's a weekend and I guess you'll book flights in advance so the date may be fixed
4. Yakushima - same reason.
Especially when you have a car I think that it's better to be able to make a spontaneous decision to stop somewhere cool that you happen to see, stay longer than planned in one place or get out of town sooner than planned, so I wouldn't reserve accommodation for other dates. YMMV.