Out of those months, I have been to LC in Sept, Oct, and Nov. I would pick Sept. every time. The island is mostly empty as most resorts are closed except LCBR, the airport on GCM is much easier to travel through, the water is generally very warm and very calm. The only downside is that with the calm winds can come very warm air temps topside during the day.
As somebody said, the closer to December you get, the better your chances are of northerly winds blowing out diving on the north side. I know for everything I am about to say, somebody will come back and say "yeah but..." and tell you they had a trip in October or November that was awesome. But my experiences in October and November were that most weeks you stand a very good chance of losing at least 1-2 days of diving on the north side and some years you will have periods of 2-3 weeks where they never get a chance to dive the north side, particularly in late October and November. Go take a look at the Cayman Aggressor's trip logs and see how many times they don't make the crossing from GCM in those months, and that is a good proxy for how often you should expect to have bad weather for diving the north side of LC. There are weeks where the weather will be awesome, as you read above, but there are also more weeks in October and November that are not awesome.
Also, a few years ago I did some poking around in the hurricane archives at the NHC for about the last 25-30 years. What I found was that while mid-September is the "peak" of hurricane season for the tropical Atlantic and Caribbean as a whole, the Caymans are far more likely to be hit by a storm coming from the southwestern Caribbean during the months of October and November. Other than hurricane Ivan, the worst storms to hit or pass close by the Cayman Islands recently have been late season storms moving from SW to NE.
When we go these days, we typically shoot for August or September and buy trip insurance just in case we have to move our flights. Last fall we were scheduled to fly through Miami the day before Irma and ended up shifting our trip by one week later. Luckily the resort had room (low season for the win!) and we were able to get a refund of our points on SW and get a cheap fare on United instead. Cayman Airways changed our flights with no fee. So while we had trip insurance, we didn't need to use it. Same story when we were supposed to fly into Little Cayman the day Ivan hit GCM in 2004. We had insurance but never had to use it.
