If you are just brining it to carry water, don't take a cooler at all.
WATER IS FREE IN THE PARKS!! Bottled water costs money, but at counter-service it's free.
Just go to any counter-type service and ask for cup(s) of ice water and they'll give it to you. We did this a ton of times and never had an issue. I'd go up by myself and ask for 4 and they'd give it to me.