At Hilton Head, if you want the best weather, it will coincide with hurricane season. However, there has not been a "direct hit" on HH in over 100 years. About 5 years ago, the entire SC/GA coastal area was evacuated, due to a hurricane warning, but that's the worst I can remember lately.
Twice we've been to HH just after a hurricane threatened the area. There were small branches down, some flooding, and the worst was that the beach house pool had to close for a couple of days because so much sand blew into it!
I wouldn't worry too much about the possibility of hurricanes, tho.
I thnk you should go for it! If you go to HH much later than late September, it may start getting too cool to really enjoy the beach.