If you are staying on or close to kiawah and need to play your second round there I would say most of the courses are fun and typically in great condition. Lots of changes lately so I would recommend based on the following:
Cougar- as of today in the best shape of “the rest of the resort” courses. Really awesome brand new club house and a couple holes on the marsh/river
Turtle- pretty standard Nicklaus course until you hit the back and there are three on the ocean. The only other kiawah course with true ocean holes
Osprey- probably my favorite just because it is fun. Not sure what kind of game you have but seems to be the most playable for all skill levels from what I’ve seen.
I’d say if your staying closer to Charleston and willing to travel that the links course at wild dunes is my second favorite “public” course in town. Ton of variation in the design from start to finish. With some significant elevation changes for the low country on the early part of the back to the ocean views on the last few holes. Really fun to play.
I am a member at stono and would say it’s the best of the “rest” of the moderately priced public courses. Couple holes on the stono river/intercoastal tons of history between the revolutionary war, a cart path that used to be the railway between savannah and Charleston and a tee box that was the pier for the ferry between johns island and west Ashley/Hollywood.
Hope you enjoy your trip!