The name they give won't (probably) have much bearing on his name as a future monarch (assuming we still have a monarchy). Charles is unlikely to be Charles III if and when he becomes king. This is a shame as I was looking forward to the day when letterboxes would have "CIII PO" written on them.
However, I reckon that they won't go for David. Besides the inanity of naming him for David Beckham, I think the Queen might be still sore about what happened with another royal named David, the one who eventually became (albeit briefly) Edward VIII. And in case you're wondering, he WAS called David at home, despite David being at the end of a ridiculously lengthy name.