We bought this one for my daughter who will be 14 months in August. She has plenty of room and it reclines. It is also taller so it doesnt put a big strain on the back.
http://www.toysrus.com/product/index.jsp?productId=28882006&cp=2255957.2273447.35373216&parentPage=family
Melbourne Beach, Indialantic Beach IMO are prettier and more fun and on the Atlantic rather than the gulf. You could also visit Dsiney's Vero Beach Resort which is only 1 hour 45 minutes from Disney. Nice resort, beautiful beach. There is a public beach right next to the resort.