You can leave your GC down near the Marina. If you are really fortunate and get a plug it can charge while you are gone.
As far as security...When you pick up your cart they will tell you that there is a $15 fee if you lose your key. SO, back in the day when before we had our own cart, we parked our rental only to return to find it gone. I went to the office and said "I still have the key, but the cart is gone!"
Apparently, it happens occasionally. They laughed and gave me a new cart. They said that they would have security look for the other one.
That was that.