I am very partial to the period from late July to the 1st week or so of August. Yes that includes the traditional peak of hurricane season in mid-September. However, I did some research on hurricane/tropical storm paths a few years ago and found that for storms passing within 100 miles or so of the Caymans, the peak was actually in late October/early November. There are of course exceptions to that such as Ivan in 2004, but in general, you are much more likely to be impacted by a storm in the Caymans in late October than in July-September.
We have been to LC something like 12 times since 1999 and all but 1 of those times was between July and mid-October. The one that wasn't was in November as a reschedule in 2004 due to Hurricane Ivan. The reasons that I like to go then are:
1 - LCBR offers special package pricing that is much cheaper than the rest of the year
2 - When there is not a hurricane around, the conditions are generally very flat with light winds and calm seas, which is good for underwater viz and the chances of diving the north side of the island where the best diving is
3 - water temps are warm and I can dive in shorts and a rash guard, with no need for a wetsuit
4 - LCBR is the only resort/dive op running for September and part of October, meaning lower #s of divers on the sites and fewer boats in the water.
5 - it is low season in general, so the flights and airports in GC are less crowded
We have had to move our vacations 3 times out of those 12 or so trips, once for a hurricane hitting the islands (Ivan in 2004) once for a tropical storm nearby, but not directly impacting GC, and once for a hurricane impacting our flight connection airport in Florida. In all three cases, LCBR moved our reservations to different dates with no fuss or fees, Cayman Airways changed our flights with no fuss or fees, and our other airlines (SW or United) changed our flights with no fuss or fees. With reputable operators like that, if you have the flexibility to change your schedule by a few weeks on short notice, you can head down there during hurricane season with no worries and no need for extra trip insurance.
In regards to some of your other desires as a non-diver, I will say that Cayman Brac may give you more of what you are looking for. CBBR has a much better pool area than LCBR and more restaurants and things to do outside of the resort areas. LCBR during low season will only have the LCBR restaurant and resort open and their pool is pretty basic, although the hammocks and bar area are pretty nice if there is a breeze. The diving on CB is better than GC, but not as good as LC in my opinion, but it is more fun for non-divers.