Whatever you plant in that basket, you're going to need to water, and in summer heat probably every day. I'd make a point of watering just before the sun hits it in the middle of the day. Give it a good soak to survive the onslaught of the bright sun.
I love the dragon wing begonias, too and they do take some sun even here in Florida. But a whole afternoon of direct sun might be too much - the leaves might burn. I'd go for the petunias or something else that will flower and grow lush. They will cascade nicely too, in fact there are varieties called "Supercascade" that are gorgeous in hanging baskets.
