We've visited many times in September. There are some restaurant closures but not many. The weather is lovely, the water very warm and fewer visitors.
Last October we spent 3 weeks. Whole different ballgame. Many, many closures which included lots of our favorites. The weather was hot and stormy. No, we didn't starve and still had a great time but we'll avoid October and probably most of November in the future. It's the slowest of the low season. But I wouldn't hesitate to go in September.