The particular direct flight is no longer available but my easiest trip ever was flying direct from CLT to LGW then immediately getting on the train (at the station in the Gatwick terminal) and heading to the southeast coast. The town of Deal specifically. After a few days of golf I hitched a ride with Tony_M back to London, short train ride back to Gatwick and stayed overnight in the Yotel in the basement of that same Gatwick terminal.
Wow, that was the quickest and most trouble-free five days of excellent links golf that I'll ever experience. A day at Princes, a day at Sandwich and three days at Royal Cinque Ports all without needing a rental car. That was a good week.