You can't use the beach club at Oil Nut but you can use the marina pool (and it's heated!) Cornhole on the beach for the kids, bar for the adults. Good coffee shop and lunch too.
Bitter End feels a bit more "grown up" now. Kids may not have much to do.
They would enjoy the Michael Beans and Jumbies shows.

And I don't care if they are chubby or stick thin, every kid who spends all day in a wet bathing suit will get the dreaded between-the-legs-vacation-ruining CHAFE. This stuff has literally saved our vacations:

https://www.bodyglide.com/product/body/

And I know this sounds silly but don't forget the SPF on their lips! My kids lips (and mine) have burned and I can tell you, it is miserable! Get each kid their own flavor of Sun Bum and make sure they use it.