We did not have any baby drinks--but we totally froze water bottles and they stayed frozen for at leat 3 -4 hours into the park--this kept everything in our bag cold---I think you can also ask for ice at the CS places. Then you could just drop it down into the cup
Lots of folks put a no spill cup of whatever in a ziploc bag with ice. As the ice melts, you can replenish it as all the CS will give you a free cup of ice.
I personally like the frozen water bottle trick best.