Kiawah is among my all time favorite destinations. For the past 10 years my wife and I have vacationed there at least 2x each year. Like most here I find TOC magical. At this time of year the heat is certainly an issue. I'd recommend an early (8 a.m.) tee time, and take bug spray. One year we christened our round "The Malaria Open." The bugs were that bad.
Unlike some others here I really do NOT enjoy Turtle Point. I find the 3 "ocean holes" particularly distasteful. They are small, narrow, short windswept holes, presenting essentially no options. To me they feel shoehorned in, so the resort could label the course "oceanfront." I do like Fazio's Osprey Point. It's quite spread out, which makes walking difficult, but the scenery and wildlife are spectacular. I enjoy Cougar, and would play it ahead of Turtle. Oak Point, just outside the front gate used to be a pretty awful course. I have not played it since the re-do, though I hear it is vastly improved.
I have also been fortunate enough to play both privates on the island several times. Cassique is really spectacular, links-style golf, and I would rate it #2 on the island, behind TOC. River is likewise beautiful, but not quite as interesting as Cassique. I haven't been back since the clubhouse burned down (tragedy, as it was quite beautiful).
I would rank the courses on Kiawah as follows:
-TOC
-Cassique (private)
-River (Private)
-Osprey
-Cougar
-Turtle Point
-Oak
I'm embarrassed to say I have not ventured off the island to play any other courses in the area. We find Kiawah Island to be a magical place, and once we're there we very rarely venture off the island. As others have said, a cocktail on the porch of the clubhouse of TOC at sunset is a wonderful way to end a day.