You can buy Perrier bottles by the case at Costco for a much better price than at regular grocery store. Might be worth seeing if you can have a case or two shipped to your resort. The Costco bottles are plastic - maybe you could tuck a few into a backpack each morning and brng to the parks with you. Just ask for a cup of ice water at CS restaurants and you're all set!
Or, try Garden Grocer or We Go Shop. You could have som delivered to your hotel and do the same thing.