I bring my cooler. No problems.
I have seen quite a few while there. Park office does sell ice as well.
Yes you can bring a cooler no problems. They ask that you clean up after yourself. I was so happy about the surroundings that I actually went with a bag and cleaned up anything I saw laying around.
I saw folks with coolers on the grass. Look at the Greenbrier's website for specific do's & don'ts.
I am not sure about having the cooler directly on the sand but you can have it on the grass that surrounds the beach. There are trees where you can do a picnic.
Thanks! Your answer is awaiting moderation.