September is not bad except for the high chance of a typhoon interfering with your holidays.
I recommend late June and early July when the weather tends to be very sunny and the school summer holidays have not started yet. Alternatively, October is very nice.
The air and water remain warm enough for swimming in October.
By the way, can you rent a car on Miyako and Ishigaki and if so, is it a lot more expensive than say Narita (since we don't speak Japanese so difficult to find English car rental websites) ?
Yes, there are lots of car rental outlets on Miyako and Ishigaki. Some of the have ENglish websites:
And here is a service that let's you make reservations through an English online form: