Brock, you should note that any golf club is hard to get on when you arrive in beach wear.
I don't know what your range of experience is with private clubs versus public courses. I doubt Savannah GC is any more or less exclusive than other clubs. Normally, it helps knowing a member, or writing a nice letter on letterhead to the pro, or havng a letter of introduction from your own club's pro or general manager. Alternatively, I I have occasionally found begging to be an effective strategy, esp. if you arrive on a Monday, go to the pro shop, get down on your hands and knees to genuflect and then buy a few shirts.
Savannah GC is nominally a Donald Ross course, though I didn't see much if anything of his legacy when I ws there in 1999. I does have some really cool Civil War-remnant bunkers and trenches on the front nine, though.